Symmetry, Integrability and Geometry: Methods and Applications
SIGMA 10 (2014), 071, 41 pages
Spherical Functions of Fundamental K-Types
Associated with the n-Dimensional Sphere
Juan Alfredo TIRAO and Ignacio Nahuel ZURRIÁN
CIEM-FaMAF, Universidad Nacional de Córdoba, Argentina
E-mail: tirao@famaf.unc.edu.ar, zurrian@famaf.unc.edu.ar
URL: http://www.famaf.unc.edu.ar/~zurrian/
Received December 20, 2013, in final form June 20, 2014; Published online July 07, 2014
http://dx.doi.org/10.3842/SIGMA.2014.071
Abstract. In this paper, we describe the irreducible spherical functions of fundamental
K-types associated with the pair (G, K) = (SO(n + 1), SO(n)) in terms of matrix hypergeometric functions. The output of this description is that the irreducible spherical functions
of the same K-fundamental type are encoded in new examples of classical sequences of
matrix-valued orthogonal polynomials, of size 2 and 3, with respect to a matrix-weight W
supported on [0, 1]. Moreover, we show that W has a second order symmetric hypergeometric
operator D.
Key words: matrix-valued spherical functions; matrix orthogonal polynomials; the matrix
hypergeometric operator; n-dimensional sphere
2010 Mathematics Subject Classification: 22E45; 33C45; 33C47
1
Introduction
The theory of spherical functions dates back to the classical papers of É. Cartan and H. Weyl;
they showed that spherical harmonics arise in a natural way from the study of functions on
the n-dimensional sphere S n = SO(n + 1)/SO(n). The first general results in this direction
were obtained in 1950 by Gel’fand, who considered zonal spherical functions of a Riemannian
symmetric space G/K. In this case we have a decomposition G = KAK. When the Abelian
subgroup A is one dimensional, the restrictions of zonal spherical functions to A can be identified with hypergeometric functions, providing a deep and fruitful connection between group
representation theory and special functions. In particular when G is compact this gives a one
to one correspondence between all zonal spherical functions of the symmetric pair (G, K) and
a sequence of orthogonal polynomials.
In light of this remarkable background it is reasonable to look for an extension of the above
results, by considering matrix-valued irreducible spherical functions on G of a general K-type.
This was accomplished for the first time in the case of the complex projective plane P2 (C) =
SU(3)/U(2) in [5]. This seminal work gave rise to a series of papers including [6, 7, 8, 10,
14, 15, 16, 17, 18, 19], where one considers matrix valued spherical functions associated to
a compact symmetric pair (G, K) of rank one, arriving at sequences of matrix valued orthogonal
polynomials of one real variable satisfying an explicit three-term recursion relation, which are
also eigenfunctions of a second order matrix differential operator (bispectral property).
The very explicit results contained in this paper are obtained for certain K-types, namely
the fundamental K-types. Also, the detailed construction of sequences of matrix orthogonal
polynomials out of these irreducible spherical functions, following the general pattern established
in [5], gives new examples of classical sequences of matrix-valued orthogonal polynomials of
size 2 and 3. For the general notions concerning matrix-valued orthogonal polynomials see [9].
2
J.A. Tirao and I.N. Zurrián
Interesting generalizations of these sequences are given in [20], where the coefficients of the three
term recursion relation satisfied by them is exhibited.
The present paper is an outgrowth of the results of [25, Chapter 5] and we are currently
working on the extension of these results for the spherical functions of any K-type associated
with the n-dimensional sphere. Using [23], one can obtain the corresponding results for the
spherical functions of any K-type associated with n-dimensional real projective space. The
starting point is to describe the irreducible spherical functions associated with the pair (G, K) =
(SO(n+1), SO(n)) in terms of eigenfunctions of a matrix linear differential operator of order two.
The output of this description is that the irreducible spherical functions of the same fundamental
K-type are encoded in a sequence of matrix valued orthogonal polynomials.
Briefly the main results of this paper are the following. After some preliminaries, in Section 3
we study the eigenfunctions of an operator ∆ on G, which is closely related to the Casimir
operator. Every spherical function Φ has to be eigenfunction of this operator ∆; considering the
KAK-decomposition
SO(n + 1) = SO(n)SO(2)SO(n)
and choosing an appropriate coordinate y on an open subset
∆Φ = λΦ, λ ∈ C, into a matrix valued differential equation
(0, 1), where H is the restriction of Φ to SO(2). The property
Φ(xgy) = π(x)Φ(g)π(y),
g ∈ G,
of A, we translate the condition
e = λH on the open interval
DH
of the spherical functions
x, y ∈ K,
tell us that Φ is determined by its K-type and the function H.
In Section 4 we first explicitly describe all the irreducible spherical functions of the symmetric
pair (G, K) = (SO(n + 1), SO(n)) with M -irreducible K-types, with M = SO(n − 1), the
centralizer of the subgroup A in K; we give these expressions in terms of the hypergeometric
function 2 F1 .
e is studied in detail when the K-types correspond to fundamental
In Section 5 the operator D
representations. Certain K-fundamental types are M irreducible, and therefore they were already considered en Section 4; besides, when n is odd there is a particular fundamental K-type
which has three M -submodules, this case is studied in the last section of this work. For the rest
of the cases we considered separately when n is even and when n is odd. Although, in both
cases we worked with the concrete realizations of the fundamental representations considering
the exterior powers of the standard representation of SO(n):
Λ1 Cn , Λ2 Cn , . . . , Λ`−1 Cn ,
with n = 2` or n = 2` + 1.
e by using the polynomial function
In Section 6 we conjugate the operator D,
2y − 1
1
Ψ(y) =
,
1
2y − 1
whose columns correspond to irreducible spherical functions, in order to obtain a matrix-valued
e
hypergeometric operator D = Ψ−1 DΨ:
DP = y(1 − y)P 00 + (C − yU )P 0 − V P,
with
(n/2 + 1)
1
C=
,
1
(n/2 + 1)
U = (n + 2)I,
p
0
V =
.
0 n−p
Spherical Functions of Fundamental K-Types Associated with the n-Dimensional Sphere
3
Then, we study all the possible eigenvalues corresponding to irreducible spherical functions
and all the polynomial eigenfunctions of D.
In Section 7, for any fundamental K-type (Λk (Cn )) with 1 ≤ p ≤ ` − 1, we find a matrixweight W , which is a scalar multiple of
2
n(2y − 1)
n/2−1 p(2y − 1) + n − p
,
W = (y(1 − y))
n(2y − 1)
(n − p)(2y − 1)2 + p
such that D is a symmetric operator with respect to the inner product defined among continuous
vector-valued functions on [0, 1] by
1
Z
hP1 , P2 iW =
P2∗ (y)W (y)P1 (y)dy.
0
Also we prove that every spherical function gives a vector polynomial eigenfunction P of D.
Therefore we obtain the following explicit expression of P in terms of the matrix hypergeometric
function for any irreducible spherical function
P (y) =
w
X
yj
j=0
j!
[C; U ; V + λ]j P (0),
see Theorem 7.6.
In Section 8 for each pair (n, p) we construct a sequence of matrix orthogonal polynomials
{Pw }w≥0 of size 2 with respect to the weight function W , which are eigenfunctions of the
symmetric differential operator D. Namely,
λ(w, 0)
0
,
DPw = Pw
0
λ(w, 1)
where
(
−w(w + n + 1) − p
if δ = 0,
λ(w, δ) =
−w(w + n + 1) − n + p if δ = 1.
Finally, in Section 9 we develop the same techniques in order to obtain analogous results for
irreducible spherical functions of the particular K-fundamental type Λ` (Cn ) for which we have
three M -submodules instead of only two. This only occurs when n is of the form 2` + 1.
It is worth to notice that, unlike the other cases, the 3 × 3 matrix-weight built here does
reduce to a smaller size.
2
2.1
Preliminaries
Spherical functions
Let G be a locally compact unimodular group and let K be a compact subgroup of G. Let K̂
denote the set of all equivalence classes of complex finite dimensional irreducible representations
of K; for each δ ∈ K̂, let ξδ denote the character of δ, d(δ) the degree of δ, i.e. the dimension
of any representation in the class Rδ, and χδ = d(δ)ξδ . We shall choose once and for all the Haar
measure dk on K normalized by K dk = 1.
We shall denote by V a finite dimensional vector space over the field C of complex numbers
and by of all linear transformations of V into V . Whenever we refer to a topology on such
a vector space we shall be talking about the unique Hausdorff linear topology on it.
4
J.A. Tirao and I.N. Zurrián
Definition 2.1. A spherical function Φ on G of type δ ∈ K̂ is a continuous function on G with
values in End(V ) such that
i) Φ(e) = I (I is the identity transformation);
R
ii) Φ(x)Φ(y) = K χδ (k −1 )Φ(xky)dk for all x, y ∈ G.
The reader can find a number of general results in [21] and [4]. For our purpose it is appropriate to recall the following facts.
Proposition 2.2 ([21, Proposition 1.2]). If Φ : G −→ End(V ) is a spherical function of type δ
then:
i) Φ(k1 gk2 ) = Φ(k1 )Φ(g)Φ(k2 ), for all k1 , k2 ∈ K, g ∈ G;
ii) k 7→ Φ(k) is a representation of K such that any irreducible subrepresentation belongs to δ.
Concerning the definition, let us point out that the spherical function Φ determines its
type univocally (Proposition 2.2) and let us say that the number of times that δ occurs in the
representation k 7→ Φ(k) is called the height of Φ.
A spherical function Φ : G −→ End(V ) is called irreducible if V has no proper subspace
invariant by Φ(g) for all g ∈ G.
If G is a connected Lie group, it is not difficult to prove that any spherical function Φ :
G −→ End(V ) is differentiable (C ∞ ), and moreover that it is analytic. Let D(G) denote the
algebra of all left invariant differential operators on G and let D(G)K denote the subalgebra of
all operators in D(G) which are invariant under all right translations by elements in K.
In the following proposition (V, π) will be a finite dimensional representation of K such that
any irreducible subrepresentation belongs to the same class δ ∈ K̂.
Proposition 2.3. A function Φ : G −→ End(V ) is a spherical function of type δ if and only if
i) Φ is analytic;
ii) Φ(k1 gk2 ) = π(k1 )Φ(g)π(k2 ), for all k1 , k2 ∈ K, g ∈ G, and Φ(e) = I;
iii) [DΦ](g) = Φ(g)[DΦ](e), for all D ∈ D(G)K , g ∈ G.
Moreover, we have that the eigenvalues [DΦ](e), D ∈ D(G)K , characterize the spherical
functions Φ as stated in the following proposition.
Proposition 2.4 ([21, Remark 4.7]). Let Φ, Ψ : G −→ End(V ) be two spherical functions on
a connected Lie group G of the same type δ ∈ K. Then Φ = Ψ if and only if (DΦ)(e) = (DΨ)(e)
for all D ∈ D(G)K .
Let us observe that if Φ : G −→ End(V ) is a spherical function, then Φ : D 7→ [DΦ](e)
maps D(G)K into EndK (V ) (EndK (V ) denotes the space of all linear maps of V into V which
commutes with π(k) for all k ∈ K) defining a finite dimensional representation of the associative
algebra D(G)K . Moreover, the spherical function is irreducible if and only if the representation
Φ : D(G)K −→ EndK (V ) is irreducible. We quote the following result from [19].
Proposition 2.5 ([19, Proposition 2.5]). Let G be a connected reductive linear Lie group. Then
the following properties are equivalent:
i) D(G)K is commutative;
ii) every irreducible spherical function of (G, K) is of height one.
Spherical Functions of Fundamental K-Types Associated with the n-Dimensional Sphere
5
In this paper the pair (G, K) is (SO(n + 1), SO(n)). Then, it is known that D(G)K is
an Abelian algebra; moreover, D(G)K is isomorphic to D(G)G ⊗ D(K)K (see in [13, Theorem 10.1] or [1]), where D(G)G (resp. D(K)K ) denotes the subalgebra of all operators in D(G)
(resp. D(K)) which are invariant under all right translations by elements in G (resp. K).
An immediate consequence of this is that all irreducible spherical functions of our pair (G, K)
are of height one.
Spherical functions of type δ (see in [21, Section 3]) arise in a natural way upon considering
representations of G. If g 7→ U (g) is a continuous representation of G, say on a finite dimensional
vector space E, then
Z
χδ k −1 U (k)dk
Pδ =
K
is a projection of E onto Pδ E = E(δ). If Pδ 6= 0 the function Φ : G −→ End(E(δ)) defined by
g ∈ G,
Φ(g)a = Pδ U (g)a,
a ∈ E(δ),
(2.1)
is a spherical function of type δ. In fact, if a ∈ E(δ) we have
Z
Φ(x)Φ(y)a = Pδ U (x)Pδ U (y)a =
χδ k −1 Pδ U (x)U (k)U (y)adk
K
Z
−1
=
χδ k
Φ(xky)dk a.
K
If the representation g 7→ U (g) is irreducible then the associated spherical function Φ is also
irreducible. Conversely, any irreducible spherical function on a compact group G arises in this
way from a finite dimensional irreducible representation of G.
2.2
Root space structure of so(n, C)
Let Eik denote the square matrix with a 1 in the ik-entry and zeros elsewhere; and let us consider
the matrices
Iki = Eik − Eki ,
1 ≤ i, k ≤ n.
Then, the set {Iki }i<k is a basis of the Lie algebra so(n). These matrices satisfy the following
commutation relations
[Iki , Irs ] = δks Iri + δri Isk + δis Ikr + δrk Iis .
If we assume that k > i, r > s then we have
[Iki , Iis ] = Isk ,
[Iki , Irk ] = Iri ,
[Iki , Iri ] = Ikr ,
[Iki , Iks ] = Iis ,
and all the other brackets are zero. From this it easily follows that the set
{Ip,p−1 : 2 ≤ p ≤ n}
generates the Lie algebra so(n).
Proposition 2.6. Given n ∈ N, we have that the operator
X
2
Qn =
Iki
∈ D(SO(n))
1≤i,k≤n
is right invariant under SO(n), i.e.
Qn ∈ D(SO(n))SO(n) ,
∀ n ∈ N0 .
6
J.A. Tirao and I.N. Zurrián
Proof . To prove that Qn is right invariant under SO(n) it is enough to prove that I˙p,p−1 (Qn ) = 0
for all 2 ≤ p ≤ n. We have
X
[Ip,p−1 , Iki ]Iki + Iki [Ip,p−1 , Iki ] .
I˙p,p−1 (Qn ) =
1≤i,k≤n
Then
I˙p,p−1 (Qn ) =
X
(Iip Ip−1,i + Ip−1,i Iip ) +
1≤i≤n
+
X
X
(Ik,p−1 Ikp + Ikp Ik,p−1 )
1≤k≤n
X
(Ipk Ik,p−1 + Ik,p−1 Ipk ) +
1≤k≤n
(Ip−1,i Ip,i + Ip,i Ip−1,i ) = 0.
1≤i≤n
This proves the proposition.
2.3
The operator Q2`
Let us assume that n = 2`. We look at a root space decomposition of so(n) in terms of the basis
elements Iki , 1 ≤ i < k ≤ n.
The linear span
h = hI21 , I43 , . . . , I2`,2`−1 iC
is a Cartan subalgebra of so(n, C). To find the root vectors it is convenient to visualize the
elements of so(n, C) as ` × ` matrices of 2 × 2 blocks. Thus h is the subspace of all diagonal
matrices of 2 × 2 skew-symmetric blocks. The subspaces of all matrices A with a block Ajk of
size two, 1 ≤ j < k ≤ `, in the place (j, k) and −Atjk in the place (k, j) with zeros in all other
places, are ad(h)-stable. Let
H = i(x1 I21 + · · · + x` I2`,2`−1 ) ∈ h,
for x1 , . . . , x` ∈ R. Then [H, A] = λ(H)A, ∀ H ∈ h, if and only if for every Ajk we have
xj (H)iI2j,2j−1 Ajk − xk (H)iAjk I2k,2k−1 = λ(H)Ajk ,
∀ H ∈ h.
Up to a scalar, the nontrivial solutions of these linear equations are the following:
1 ±i
Ajk =
with corresponding λ = ∓(xj + xk ),
±i −1
1 ∓i
Ajk =
with corresponding λ = ∓(xj − xk ).
±i 1
Let j ∈ h∗ be defined by j (H) = xj for 1 ≤ j ≤ `. Then for 1 ≤ j < k ≤ `, the following
matrices are root vectors of so(2`, C):
Xj +k = I2k−1,2j−1 − I2k,2j − i(I2k−1,2j + I2k,2j−1 ),
X−j −k = I2k−1,2j−1 − I2k,2j + i(I2k−1,2j + I2k,2j−1 ),
Xj −k = I2k−1,2j−1 + I2k,2j − i(I2k−1,2j − I2k,2j−1 ),
X−j +k = I2k−1,2j−1 + I2k,2j + i(I2k−1,2j − I2k,2j−1 ).
Thus, if we choose the following set of positive roots
∆+ = {j + k , j − k : 1 ≤ j < k ≤ `},
(2.2)
Spherical Functions of Fundamental K-Types Associated with the n-Dimensional Sphere
7
then the Dynkin diagram of so(2`, C) is D` :
◦
...
◦
◦
1 − 2
2 − 3
`−1 − `
◦
@
`−2 − `−1
@◦
`−1 + `
By looking at the
1
Xj +k =
−i
1
Xj −k =
−i
2 × 2 blocks Ajk of the different roots, namely
1 i
−i
,
,
X−j −k =
i −1
−1
1 −i
i
,
,
X−j +k =
i 1
1
it is easy to obtain the following inverse relations
1
4
Xj +k + X−j −k + Xj −k + X−j +k ,
I2k,2j = 41 − Xj +k − X−j −k + Xj −k + X−j +k ,
I2k,2j−1 = 4i Xj +k − X−j −k − Xj −k + X−j +k ,
I2k−1,2j = 4i Xj +k − X−j −k + Xj −k − X−j +k .
I2k−1,2j−1 =
From this it follows that
2
2
2
2
I2k−1,2j−1
+ I2k,2j
+ I2k,2j−1
+ I2k−1,2j
=
1
4
Xj +k X−j −k + X−j −k Xj +k + Xj −k X−j +k + X−j +k Xj −k .
Therefore
Q2` =
X
2
I2j,2j−1
+
X
1
4
Xj +k X−j −k + X−j −k Xj +k
1≤j<k≤`
1≤j≤`
+ Xj −k X−j +k + X−j +k Xj −k .
Now using the expressions in (2.2) we get
[Xj +k , X−j −k ] = −4i(I2j,2j−1 + I2k,2k−1 ),
[Xj −k , X−j +k ] = −4i(I2j,2j−1 − I2k,2k−1 ).
Thus Q2` becomes
X
X
2
Q2` =
I2j,2j−1
−2
(` − j)iI2j,2j−1
1≤j≤`
+
X
1≤j≤`
1
2
X−j −k Xj +k + X−j +k Xj −k .
(2.3)
1≤j<k≤`
2.4
The operator Q2`+1
Now we look at a root space decomposition of so(n) in terms of the basis elements Iki , 1 ≤ i <
k ≤ n when n = 2` + 1.
The linear span
h = hI21 , I43 , . . . , I2`,2`−1 iC
is a Cartan subalgebra of so(n, C). To find the root vectors it is convenient to visualize the
elements of so(n, C) as ` × ` matrices of 2 × 2 blocks occupying the left upper corner of the
8
J.A. Tirao and I.N. Zurrián
square matrices of size 2` + 1, with the last column (respectively row) made up of ` columns
(respectively rows) of size two and a zero in the place (2` + 1, 2` + 1). The subspaces of all
matrices A with a block Ajk , 1 ≤ j < k ≤ `, in the place (j, k), with the block −Atjk in the place
(k, j) and with zeros in all other places, are ad(h)-stable. Also the subspaces of all matrices B
with a column Bj of size two, 1 ≤ j ≤ `, in the place (j, ` + 1), with the row −Bjt in the place
(` + 1, j) and with zeros in all other places, are ad(h)-stable.
On the other hand [H, B] = λB if and only if
xj iI2j,2j−1 Bj = λBj .
Up to a scalar this linear equation has two linearly independent solutions:
1
Bj =
with corresponding λ = ∓xj ,
±i
Let ∈ h∗ be defined by (H) = xj for 1 ≤ j ≤ `. Then for 1 ≤ j < k ≤ ` and 1 ≤ r ≤ `, the
following matrices are root vectors of so(2` + 1, C):
Xj +k = I2k−1,2j−1 − I2k,2j − i(I2k−1,2j + I2k,2j−1 ),
X−j −k = I2k−1,2j−1 − I2k,2j + i(I2k−1,2j + I2k,2j−1 ),
Xj −k = I2k−1,2j−1 + I2k,2j − i(I2k−1,2j − I2k,2j−1 ),
X−j +k = I2k−1,2j−1 + I2k,2j + i(I2k−1,2j − I2k,2j−1 ),
Xr = In,2r−1 − iIn,2r ,
X−r = In,2r−1 + iIn,2r .
Thus, if we choose the following set of positive roots
∆+ = {r , j + k , j − k : 1 ≤ r ≤ `, 1 ≤ j < k ≤ `},
then the Dynkin diagram of so(2` + 1, C) is B` :
...
◦
◦
1 − 2
2 − 3
◦
>◦
`−1 − `
`
By looking at the 2 × 1 columns of the different roots, namely
1
1
Xj =
,
X−j =
,
i
−i
it is easy to obtain the following inverse relations
In,2r−1 = 21 (Xr + X−r ),
In,2r = 2i (Xr − X−r ).
From this it follows that
2
2
In,2r−1
+ In,2r
= 21 (Xr X−r + X−r Xr ) = −iI2r,2r−1 + X−r Xr ,
since [Xr , X−r ] = −2iI2r,2r−1 . Therefore we have that
X
X
2
Q2`+1 =
In,j
+ Q2` =
(−iI2r,2r−1 + X−r Xr ) + Q2` .
1≤j≤2`
1≤r≤2`
Then
Q2`+1 =
X
2
I2j,2j−1
−
1≤j≤`
+
X
1≤j<k≤`
X
(2(` − j) + 1)iI2j,2j−1
1≤j≤`
1
2
X
X−j −k Xj +k + X−j +k Xj −k +
X−r Xr .
1≤r≤2`
(2.4)
Spherical Functions of Fundamental K-Types Associated with the n-Dimensional Sphere
2.5
9
Gel’fand–Tsetlin basis
For any n we identify the group SO(n) with a subgroup of SO(n + 1) in the following way: given
k ∈ SO(n) we have
k 0
k'
∈ SO(n + 1).
0 1
Let Tm be an irreducible unitary representation of SO(n) with highest weight m and let Vm
be the space of this representation. Highest weights m of these representations are given by
the `-tuples of integers m = mn = (m1n , . . . , m`n ) for which
m1n ≥ m2n ≥ · · · ≥ m`−1,n ≥ |m`n |
m1n ≥ m2n ≥ · · · ≥ m`n ≥ 0
if n = 2`,
if n = 2` + 1,
and mjn are all integers.
The restriction of the representation Tm of the group SO(2` + 1) to the subgroup SO(2`)
decomposes into the direct sum of all representations Tm0 , m0 = mn−1 = (m1,n−1 , . . . , m`,n−1 )
for which the betweenness conditions
m1,2`+1 ≥ m1,2` ≥ m2,2`+1 ≥ m2,2` ≥ · · · ≥ m`,2`+1 ≥ m`,2` ≥ −m`,2`+1
are satisfied. For the restriction of the representations Tm of SO(2`) to the subgroup SO(2` − 1)
the corresponding betweenness conditions are
m1,2` ≥ m1,2`−1 ≥ m2,2` ≥ m2,2`−1 ≥ · · · ≥ m`−1,2` ≥ m`−1,2`−1 ≥ |m`,2` |.
All multiplicities in the decompositions are equal to one (see [24, p. 362]).
If we continue this procedure of restriction of irreducible representations successively to the
subgroups
SO(n − 2) > SO(n − 3) > · · · > SO(2),
then we finally get one dimensional representations of the group SO(2). If we take a unit
vector in each one of these one dimensional representations we get an orthonormal basis of
the representation space Vm . Such a basis is called a Gel’fand–Tsetlin basis. The elements of
a Gel’fand–Tsetlin basis {v(µ)} of the representation Tm of SO(n) are labelled by the Gel’fand–
Tsetlin patterns µ = (mn , mn−1 , . . . , m3 , m2 ), where the betweenness conditions are depicted
in the following diagrams.
If n = 2` + 1

m1n
m2n
m1,n−1





µ=





−m` n
m` n
m`,n−1
m15
−m25
m25
m14
m24
−m13
m13
m12






.





If n = 2`






µ=





m1n
m2n
m` n
m1,n−1
m`−1,n−1
m15
m25
m14
m24
m13
m12


−m`−1,n−1 



.
−m25




−m13

10
J.A. Tirao and I.N. Zurrián
The chain of subgroups SO(n − 1) > SO(n − 2) > · · · > SO(2) defines the orthonormal basis
{v(µ)} uniquely up to multiplication of the basis elements by complex numbers of absolute value
one.
2.6
An explicit expression for π̇(Qn )
ˆ
Since Qn ∈ D(SO(n))SO(n) , given π̇ ∈ SO(n)
it follows that π̇(Qn ) commutes with π(k) for all
k ∈ SO(n). Hence, by Schur’s Lemma π̇(Qn ) = λI. From expressions (2.3) and (2.4) we can
give the explicit value of λ in terms of the highest weight of π, by computing π̇(Qn ) on a highest
weight vector.
Proposition 2.7. Let (π, Vπ ) be an irreducible representation of SO(2`) of highest weight m =
(m1 , m2 , . . . , m` ). Then, π̇(Q2` ) = λI, with
X
λ=
−m2j − 2(` − j)mj .
(2.5)
1≤j≤`
Proposition 2.8. Let (π, Vπ ) be an irreducible representation of SO(2` + 1) of highest weight
m = (m1 , m2 , . . . , m` ). Then, π̇(Q2`+1 ) = λI, with
X
λ=
−m2j − (2(` − j) + 1)mj .
(2.6)
1≤j≤`
3
The dif ferential operator ∆
We shall look closely at the left invariant differential operator ∆ of SO(n + 1) defined by
∆=
n
X
2
In+1,j
,
j=1
in order to study its eigenfunctions and eigenvalues. Later we will use all this to understand
the irreducible spherical functions of fundamental K-types associated with the pair (G, K) =
(SO(n + 1), SO(n)).
Proposition 3.1. Let G = SO(n + 1) and K = SO(n). Let us consider the following left
invariant differential operator of G
∆=
n
X
2
In+1,j
.
j=1
Then ∆ is also right invariant under K.
Proof . This is a direct consequence of the identity
Qn+1 = Qn + ∆
and Proposition 2.6.
Let us define the one-parameter subgroup A of G as the set of all elements of the form


In−1
0
0
cos s sin s  ,
a(s) =  0
−π ≤ s ≤ π,
(3.1)
0
− sin s cos s
Spherical Functions of Fundamental K-Types Associated with the n-Dimensional Sphere
11
where In−1 denotes the identity matrix of size n − 1, and let M = SO(n − 1) be the centralizer
of A in K.
Now we want to get the expression of [∆Φ](a(s)) for any smooth function Φ on G with values
in End(Vπ ) such that Φ(kgk 0 ) = π(k)Φ(g)π(k 0 ) for all g ∈ G and all k, k 0 ∈ K.
We have
2
∂2
In+1,j Φ (a(s)) = 2 Φ(a(s) exp tIn+1,j )
∂t
.
t=0
Hence, we will use the decomposition G = KAK to write a(s) exp tIn+1,j = k(s, t)a(s, t)h(s, t),
with k(s, t), h(s, t) ∈ K and a(s, t) ∈ A.
Let us take on A \ {a(π)} the coordinate function x(a(s)) = s, with −π < s < π, and let
F (s) = F (x(a(s))) = Φ(a(s)).
From now on we will assume that −π < s, t, s + t < π.
If j = n we have a(s) exp tIn+1,n = a(s)a(t) = a(s + t). Thus we may take
a(s, t) = a(s + t),
k(s, t) = h(s, t) = e.
Since x(a(s + t)) = s + t, we obtain
2
∂2
In+1,n Φ (a(s)) = 2 Φ(a(s) exp tIn+1,n )
∂t
=
∂2
∂t2
=
t=0
∂2
Φ(a(s + t))
∂t2
= F 00 (s).
F (s + t)
t=0
For 1 ≤ j ≤ n − 1, when s ∈
/ Zπ, we may take


Ij−1
0
0
0
0
sin t
 0
√ sin s cos t
√
0
0


1−cos2 s cos2 t
1−cos2 s cos2 t


0
In−j−1
0
0 ,
k(s, t) =  0


√ − sin t
√ sin s cos t
0
0
 0
1−cos2 s cos2 t
1−cos2 s cos2 t
0
0
0
0
1


Ij−1
0
0
0
0
sin s
 0
√
√ − cos s sin t
0
0


1−cos2 s cos2 t
1−cos2 s cos2 t


0
In−j−1
0
0 ,
h(s, t) =  0


sin s
√ cos s sin t
√
0
0
 0
1−cos2 s cos2 t
1−cos2 s cos2 t
0
0
0
0
1


In−1
0
0
√

0
1 − cos2 s cos2 t .
a(s, t) =
√ cos s cos t
2
2
0
− 1 − cos s cos t
cos s cos t
Then, for 0 < s < π, we have x(a(s, t)) = arccos(cos s cos t) and
∂
cos s sin t
x(a(s, t)) = √
.
∂t
1 − cos2 s cos2 t
From here we get
∂
x(a(s, t))
∂t
=0
t=0
and
∂2
x(a(s, t))
∂t2
=
t=0
cos s
.
sin s
t=0
12
J.A. Tirao and I.N. Zurrián
Thus
∂
Φ(a(s, t))
∂t
= F 0 (s)
t=0
∂
x(a(s, t))
∂t
=0
∂2
Φ(a(s, t))
∂t2
and
t=0
=
t=0
cos s 0
F (s).
sin s
We observe that k(s, 0) = h(s, 0) = e and that a(s, 0) = a(s). Then
2
[Inj
Φ](a(s)) =
∂2
∂
∂
π(k(s, t))
Φ(a(s)) + 2 π(k(s, t))
Φ(a(s, t))
2
∂t
∂t
t=0
t=0 ∂t
t=0
∂
∂2
∂
Φ(a(s)) π(h(s, t))
+ 2 Φ(a(s, t))
+ 2 π(k(s, t))
∂t
∂t
∂t
t=0
t=o
t=0
∂
∂
∂2
+ 2 Φ(a(s, t))
+ Φ(a(s)) 2 π(h(s, t))
π(h(s, t))
.
∂t
∂t
t=0 ∂t
t=0
t=0
We also have
∂
π (k(s, t))
∂t
t=0
= π̇
∂
k(s, t)
∂t
t=0
=
1
π̇(In,j ),
sin s
and
∂
π(h(s, t))
∂t
t=0
= π̇
∂
h(s, t)
∂t
t=0
=−
cos s
π̇ (In,j ) .
sin s
We will need the following proposition, whose proof appears in the Appendix and its idea is
taken from [5].
Proposition 3.2. If A(s, t) = k(s, t) or A(s, t) = h(s, t), then in either case for 0 < s < π, we
have
2
∂ 2 (π ◦ A)
∂A
= π̇
.
∂t2
∂t t=0
t=0
Moreover in each case, for 1 ≤ j ≤ n − 1 and 0 < s < π, we have
∂2
π(k(s, t))
∂t2
1
∂2
2
π̇(I
)
,
π(h(s, t))
n,j
∂t2
t=0
sin2 s
Now we obtain the following corollaries.
=
t=0
=
cos2 s
π̇(In,j )2 .
sin2 s
Corollary 3.3. Let Φ be any smooth function on G with values in End(Vπ ) such that Φ(kgk 0 ) =
π(k)Φ(g)π(k 0 ) for all g ∈ G and all k, k 0 ∈ K. Then, if F (s) = Φ(a(s)), for 0 < s < π we have
n−1
[∆Φ](a(s)) = F 00 (s) + (n − 1)
−2
cos s 0
1 X
F (s) +
π̇(In,j )2 F (s)
sin s
sin2 s j=1
n−1
n−1
X
cos s X
cos2 s
π̇(I
)F
(s)
π̇(I
)
+
F
(s)
π̇(In,j )2 .
n,j
n,j
2
2
sin s j=1
sin s
j=1
Corollary 3.4. Let Φ be an irreducible spherical function on G of type π ∈ K̂. Then, if
F (s) = Φ(a(s)), we have
n−1
F 00 (s) + (n − 1)
−2
cos s 0
1 X
F (s) +
π̇(In,j )2 F (s)
sin s
sin2 s j=1
n−1
n−1
X
cos s X
cos2 s
π̇(I
)F
(s)
π̇(I
)
+
F
(s)
π̇(In,j )2 = λF (s),
n,j
n,j
sin2 s j=1
sin2 s
j=1
for some λ ∈ C and 0 < s < π.
Spherical Functions of Fundamental K-Types Associated with the n-Dimensional Sphere
13
Notice that the expression in Corollary 3.4 generalizes the very well known situation when the
K-type is the trivial one, as we state in the following corollary (cf. [11, p. 403, equation (10)]).
Corollary 3.5. Let Φ be an irreducible spherical function on G of the trivial K-type. Then, for
F (s) = Φ(a(s)) we have
cos s 0
F 00 (s) + (n − 1)
F (s) = λF (s),
sin s
for some λ ∈ C and 0 < s < π.
Let us make the change of variables y = (1 + cos s)/2, with 0 < s < π; then 0 < y < 1. We
d
also have cos s = 2y − 1, sin2 s = 4y(1 − y) and dy
= − sin2 s . If we let H(y) = F (s), i.e.
H(y) = Φ(a(s)),
with
cos s = 2y − 1,
we obtain
sin s 0
sin2 s 00
cos s 0
H (s),
F 00 (s) =
H (y) −
H (y).
2
4
2
In terms of this new variable Corollary 3.4 becomes
F 0 (s) = −
Corollary 3.6. Let Φ be an irreducible spherical function on G of type π ∈ K̂. Then, if
H(y) = Φ(a(s)) with y = (1 + cos s)/2, we have
n−1
X
1
1
π̇(In,j )2 H(y)
y(1 − y)H 00 (y) + n(1 − 2y)H 0 (y) +
2
4y(1 − y)
j=1
+
n−1
n−1
j=1
j=1
X
(1 − 2y) X
(1 − 2y)2
H(y)
π̇(In,j )H(y)π̇(In,j ) +
π̇(In,j )2 = λH(y),
2y(1 − y)
4y(1 − y)
for some λ ∈ C and 0 < y < 1.
Remark 3.7. Let us notice that, for any y ∈ (0, 1), H(y) is a scalar linear transformation
when restricted to any M -submodule, see Proposition 2.2. Therefore, if m is the number of M submodules contained in (V, π), we consider the vector valued function H : (0, 1) → Cm whose
entries are given by those scalar values that H(y) takes on every M -submodule.
If the End(V )-valued function H satisfies the differential equation given in Corollary 3.6,
then the vector valued function H satisfies
1
1
y(1 − y)H 00 (y) + n(1 − 2y)H 0 (y) +
N1 H(y)
2
4y(1 − y)
(1 − 2y)
(1 − 2y)2
+
EH(y) +
N2 H(y) = λH(y),
2y(1 − y)
4y(1 − y)
where E, N1 and N2 are matrices of size m × m.
n−1
n−1
P 2
P 2
Even more, since
In,j = Qn − Qn−1 , Proposition 2.6 implies
In,j ∈ D(SO(n))SO(n−1) ,
j=1
therefore
n−1
P
j=1
π̇(In,j )2 is scalar valued when restricted to any M -submodule. Hence, N1 = N2
j=1
and the equation above is equivalent to
(1 − 2y)
1 + (1 − 2y)2
n
y(1 − y)H 00 (y)+ (1 − 2y)H 0 (y) +
EH(y) +
N H(y) = λH(y), (3.2)
2
2y(1 − y)
4y(1 − y)
where N is a diagonal matrix of size m × m. To obtain an explicit expression of E for any
K-type is a very serious matter; in the following sections we shall find explicitly the expressions
of E and N , for certain K-types.
14
J.A. Tirao and I.N. Zurrián
Remark 3.8. It is worth to observe that from (2.5) and (2.6) we can immediately obtain every
entry of the diagonal matrix N .
4
The K-types which are M -irreducible
Let K = SO(n), M = SO(n−1), with n = 2`+1, and let mn = (m1n , . . . , m`n ) be a K-type such
that Vm is irreducible as M -module. The highest weights mn−1 of the M -submodules of Vm are
those that satisfies the following intertwining relations
m1n
m2n
m1,n−1
...
...
−m`n
m`,n
...
m`,n−1
.
Since Vm is irreducible as M -module it follows that m1n = · · · = m`,n = 0. The converse is
also true, therefore Vm is M -irreducible if and only if it is the trivial representation.
Let now consider the case K = SO(n), M = SO(n − 1), with n = 2` and let mn =
(m1n , . . . , m`n ) be a K-type such that Vm is irreducible as M -module. The highest weights
mn−1 of the M -submodules of Vm are those that satisfies the following intertwining relations
m1n
m2n
m1,n−1
...
...
m`−1,n
...
m`n
m`−1,n−1
−m`−1,n−1
.
Since Vm is irreducible as M -module it follows that m1n = · · · = m`−1,n = d and m`n = d − j
with 0 ≤ j ≤ 2d, since m`−1,n ≥ |m`n |. This implies that m1,n−1 = · · · = m`−2,n−1 = d and
m`−1,n−1 = q with d ≥ q ≥ max{d − j, j − d}. Thus, if 0 ≤ j ≤ d we have d ≥ q ≥ d − j and
by irreducibility we must have j = 0. Similarly if d ≤ j ≤ 2d we have d ≥ q ≥ j − d and by
irreducibility we must have j = 2d. Therefore mn = dα or mn = dβ, where
α = (1, . . . , 1),
β = (1, . . . , 1, −1).
The converse is also true, therefore Vm is M -irreducible if and only if mn = dα or mn = dβ
for any d ∈ N0 .
If Φ is an irreducible spherical function on SO(n + 1) of type π, whose highest weight is
mn = dα or mn = dβ, then from Corollary 3.6 we get that the associated function H satisfies
n−1
1−y X
y(1 − y)H (y) + `(1 − 2y)H (y) +
π̇(Inj )2 H(y) = λH(y).
y
00
0
j=1
To compute
n−1
P
π̇(Inj )2 we write
j=1
n−1
P
π̇(Inj )2 = π̇(Qn − Qn−1 ).
j=1
Let us first consider mn = dα. If v ∈ Vmn is a highest weight vector, then
π̇(Qn )v = −d`(d + ` − 1)v
and
π̇(Qn−1 )v = −d(` − 1)(d + ` − 1)v,
see (2.5) and (2.6). Therefore
n−1
X
π̇(Inj )2 v = −d(d + ` − 1)v.
j=1
Let us now consider mn = dβ. If v ∈ Vmn is a highest weight vector, then π̇(Qn )v =
−2d`(d + ` − 1)v as before, and π̇(Qn−1 )v = −2d(` − 1)(d + ` − 1)v as before because in both
cases mn−1 is the same.
Spherical Functions of Fundamental K-Types Associated with the n-Dimensional Sphere
15
Therefore if mn = (d, . . . , d, ±d) we have
n−1
X
π̇(Inj )2 v = −d(d + ` − 1)v.
j=1
Hence, if Φ is an irreducible spherical function on SO(n + 1), n = 2`, of type mn =
(d, . . . , d, ±d) ∈ C` , then the associated scalar value function H = h satisfies
y(1 − y)h00 (y) + `(1 − 2y)h0 (y) −
d(d + ` − 1)(1 − y)
h(y) = λh(y).
y
(4.1)
Let us now compute the eigenvalue λ corresponding to the spherical function of type π ∈
ˆ
SO(2`), of highest weight mn = dα, associated with the irreducible representation τ ∈ SO(2`+1),
of highest weight mn+1 = (w, d, . . . , d) ∈ C` . If v ∈ Vmn+1 is a highest weight vector, then
from (2.6) we have
τ̇ (Qn+1 )v = − (w(w + 2` − 1) + d(` − 1)(d + ` − 1)) v.
If v ∈ Vmn is a highest weight vector, then from (2.5) we have
τ̇ (Qn )v = π̇(Qn )v = −d`(d + ` − 1)v.
Since ∆ = Qn+1 − Qn it follows that
λ = −w(w + 2` − 1) + d(d + ` − 1).
To solve (4.1) we write h = y α f . Then we get
y(1 − y)y α f 00 + (2α(1 − y) + `(1 − 2y))y α f 0
+ (α(α − 1)(1 − y) + `α(1 − 2y) − d(d + ` − 1)(1 − y))y α−1 f = λy α f.
Thus the indicial equation is α(α − 1) + `α − d(d + ` − 1) = 0 and α = d is one of its solutions.
If we take h = y d f , then we obtain
y(1 − y)f 00 + (2d + ` − 2(d + `)y)f 0 − d`f = λf.
If we replace λ = −w(w + 2` − 1) + d(d + ` − 1) we get
y(1 − y)f 00 + (2d + ` − 2(d + `)y)f 0 − (d − w)(2` + d + w − 1)f = 0.
Let a = d − w, b = 2` + d + w − 1, c = 2d + ` then the above equation becomes
y(1 − y)f 00 + (c − (1 + a + b)y)f 0 − abf = 0.
A fundamental system of solutions of this equation near y = 0 is given by the following
functions
a, b
a − c + 1, b − c + 1
1−c
F
;
y
,
y
F
;
y
.
2 1
2 1
c
2−c
Since h = y d f is bounded near y = 0 it follows that
d − w, 2` + d + w − 1
d
h(y) = uy 2 F1
;y ,
2d + `
where the constant u is determined by the condition h(1) = 1.
16
J.A. Tirao and I.N. Zurrián
Remark 4.1. Let hw = hw (y), w ≥ d, be the function h above. Then hw is a polynomial of
degree w. Moreover observe that the function y d used to hypergeometrize (4.1) is precisely hd .
Let us now compute the eigenvalue λ corresponding to the spherical function of type mn =
dβ associated with an irreducible representation τ of SO(n + 1) of highest weight mn+1 =
(w, d, . . . , d) ∈ C` . If v ∈ Vmn+1 is a highest weight vector, we obtain τ̇ (Qn+1 )v = −(w(w + 2` −
1) + d(` − 1)(d + ` − 1))v.
If v ∈ Vmn is a highest weight vector, then π̇(Qn )v = −d`(d + ` − 1)v as above, because Qn v
does not depend on the sign of the last coordinate of mn . Since ∆ = Qn+1 − Qn we also have
λ = −w(w + 2` − 1) + d(d + ` − 1).
Therefore we have proved the following result.
Theorem 4.2. The scalar valued functions H = h associated with the irreducible spherical
functions on SO(n + 1), n = 2`, of SO(n)-type mn = (d, . . . , d, ±d) ∈ C` , are parameterized by
the integers w ≥ d and are given by
d − w, 2` + d + w − 1
d
hw (y) = uy 2 F1
;y
2d + `
where the constant u is determined by the condition hw (1) = 1.
5
The operator ∆ for fundamental K-types
We are interested in finding a more explicit expression of the differential equation given in
Corollary 3.6:
n−1
X
1
1
y(1 − y)H 00 (y) + n(1 − 2y)H 0 (y) +
π̇(In,j )2 H(y)
2
4y(1 − y)
j=1
+
(1 − 2y)
2y(1 − y)
n−1
X
n−1
π̇(In,j )H(y)π̇(In,j ) +
j=1
X
(1 − 2y)2
H(y)
π̇(In,j )2 = λH(y),
4y(1 − y)
j=1
ˆ
for certain representations π ∈ SO(n),
including those that are fundamental.
The obvious place to start to look for irreducible representations of SO(n) is among the
exterior powers of the standard representation of SO(n). It is known that Λp (C2` ) are irreducible
SO(2`)-modules for p = 1, . . . , `−1, and that Λ` (C2` ) splits into the direct sum of two irreducible
submodules. While in the odd case Λp (C2`+1 ) are irreducible SO(2`+1)-modules for p = 1, . . . , `.
See Theorems 19.2 and 19.14 in [3].
Moreover, Λp (Cn ) and Λn−p (Cn ) are isomorphic SO(n)-modules. In fact, if {e1 , . . . , en } is
the canonical basis of Cn , then the linear map ξ : Λp (Cn ) → Λn−p (Cn ) defined by
ξ(eu1 ∧ · · · ∧ eup ) = (−1)u1 +···+up ev1 ∧ · · · ∧ evn−p ,
where u1 < · · · < up and v1 < · · · < vn−p are complementary ordered set of indices, is an
SO(n)-isomorphism.
All these statements can be established directly upon observing that the elements Iki =
Eki − Eik with 1 ≤ i < k ≤ n form a basis of the Lie algebra so(n), and that
Iki ek = ei ,
Iki ei = −ek
and
Iki ej = 0
if j 6= k, i.
We will refer to the irreducible SO(2`)-modules Λp (C2` ) for p = 1, . . . , ` − 1, respectively, the
irreducible SO(2` + 1)-modules Λp (C2`+1 ) for p = 1, . . . , `, as the fundamental SO(2`)-modules,
respectively, as the fundamental SO(2` + 1)-modules, for reasons that will be clarified in the
following Sections 5.1 and 5.2.
Spherical Functions of Fundamental K-Types Associated with the n-Dimensional Sphere
5.1
17
The even case: K = SO(2`)
First we will study the case n = 2`, with ` > 2. The fundamental weights of so(2`, C) are
λp = 1 + · · · + p ,
λ`−1 =
1
2 (1
1 ≤ p ≤ ` − 2,
+ · · · + `−1 − ` ),
λ` = 12 (1 + · · · + `−1 + ` ).
Here we will consider the fundamental K-modules
Λ1 Cn , Λ2 Cn , . . . , Λ`−1 Cn .
We will show that the highest weight of Λp (Cn ) is 1 + · · · + p for 1 ≤ p ≤ ` − 1. Observe that
λ`−1 and λ` are not analytically integral and therefore they will not be considered, although we
will also consider the K-module with highest weight λ`−1 + λ` = 1 + · · · + `−1 . Notice that we
have already considered the cases 2λ`−1 and 2λ` in Section 4, which are M -irreducible. We will
also show that the fundamental K-modules are direct sum of two irreducible M -submodules.
In order to obtain the explicit expression of E in (3.2) for a given irreducible representation π
of K = SO(n), of highest weight ε1 + · · · + εp , we are interested to compute
n−1
X
π̇(Inj )Ps π̇(Inj )
Vr
= λ(r, s)IVr ,
j=1
with r, s = 0, 1 corresponding to the two M -submodules V0 and V1 of the representation π,
associated with mn−1 = (1, . . . , 1, 0, . . . , 0) ∈ C`−1 with p − 1 and p ones, respectively (see the
betweenness conditions in Section 2.5); being P0 and P1 the respective projections.
Let us consider the standard action of K = SO(n) on V = Cn , and take the canonical basis
{e1 , . . . , en }. Then we have the irreducible K-module Λp (V ) for 1 ≤ p ≤ ` − 1. The vector
(e1 − ie2 ) ∧ (e3 − ie4 ) ∧ · · · ∧ (e2p−1 − ie2p ) is the unique, up to a scalar, dominant vector and
its weight is (1, . . . , 1, 0, . . . , 0) ∈ C` with p ones. Then, if V 0 is the subspace generated by
{e1 , . . . , en−1 }, Λp (V ) is the direct sum of two M -submodules, namely
Λp (V ) = V0 ⊕ V1 = Λp−1 (V 0 ) ∧ en ⊕ Λp (V 0 )
(5.1)
whose highest weights are (1, . . . , 1, 0, . . . , 0) ∈ C`−1 with p−1 ones and (1, . . . , 1, 0, . . . , 0) ∈ C`−1
with p ones, respectively. It is easy to see that (e1 − ie2 ) ∧ (e3 − ie4 ) ∧ · · · ∧ (e2p−3 − ie2p−2 ) ∧ en is
an M -highest weight vector in Λp−1 (V 0 ) ∧ en and that (e1 − ie2 ) ∧ (e3 − ie4 ) ∧ · · · ∧ (e2p−1 − ie2p )
is an M highest weight vector in Λp (V 0 ).
To get λ(0, 0) it is enough to compute
n−1
X
π̇(Inj )P0 π̇(Inj )(e1 ∧ · · · ∧ ep−1 ∧ en ).
j=1
Since we have that π̇(Inj )(e1 ∧ · · · ∧ ep−1 ∧ en ) = e1 ∧ · · · ∧ ep−1 ∧ ej we obtain P0 π̇(Inj )(e1 ∧
· · · ∧ ep−1 ∧ en ) = 0 and λ(0, 0) = 0.
To get λ(0, 1) it is enough to compute
n−1
X
π̇(Inj )P1 π̇(Inj )(e1 ∧ · · · ∧ ep−1 ∧ en ).
j=1
We have
(
0
P1 π̇(Inj )(e1 ∧ · · · ∧ ep−1 ∧ en ) =
e1 ∧ · · · ∧ ep−1 ∧ ej
if 1 ≤ j ≤ p − 1,
if p ≤ j ≤ n − 1.
18
J.A. Tirao and I.N. Zurrián
Therefore we have
(
0
π̇(Inj )P1 π̇(Inj )(e1 ∧ · · · ∧ ep−1 ∧ en ) =
−e1 ∧ · · · ∧ ep−1 ∧ en
if 1 ≤ j ≤ p − 1,
if p ≤ j ≤ n − 1.
Hence λ(0, 1) = −(n − p).
Similarly, to get λ(1, 0) it is enough to compute
n−1
X
π̇(Inj )P0 π̇(Inj )(e1 ∧ · · · ∧ ep ).
j=1
We have
(
−e1 ∧ · · · ∧ en ∧ · · · ∧ ep
π̇(Inj )(e1 ∧ · · · ∧ ep ) =
0
where en appears in the j-place. Therefore
(
−e1 ∧ · · · ∧ ep
π̇(Inj )P0 π̇(Inj )(e1 ∧ · · · ∧ ep ) =
0
if 1 ≤ j ≤ p,
if p + 1 ≤ j ≤ n − 1,
if 1 ≤ j ≤ p,
if p + 1 ≤ j ≤ n − 1.
Hence λ(1, 0) = −p.
Also it is clear now that
n−1
P
π̇(Inj )P1 π̇(Inj )(e1 ∧ · · · ∧ ep ) = 0, hence λ(1, 1) = 0.
j=1
Therefore, when π is the standard representation of K in Λp (V ), 1 ≤ p ≤ ` − 1, we have
0 p−n
.
(λ(r, s))0≤r,s≤1 =
−p
0
Therefore, we obtain a more explicit version of Corollary 3.6 using (3.2) and Remark 3.8.
ˆ
Corollary 5.1. Let Φ be an irreducible spherical function on G of type π ∈ SO(n),
n = 2`. If
`
the highest weight of π is of the form (1, . . . , 1, 0, . . . , 0) ∈ C , with p ones, 1 ≤ p ≤ ` − 1, then
the function H : (0, 1) → End(C2 ) associated with Φ satisfies
1
1 + (1 − 2y)2 p − n 0
00
0
y(1 − y)H (y) + n(1 − 2y)H (y) +
H(y)
0
−p
2
4y(1 − y)
(1 − 2y)
0 p−n
H(y) = λH(y),
+
0
2y(1 − y) −p
for some λ ∈ C.
5.2
The odd case: K = SO(2` + 1)
We now study the case n = 2` + 1, with ` ≥ 1. The fundamental weights of so(2` + 1, C) are
λp = 1 + · · · + p ,
λ` =
1
2 (1
1 ≤ p ≤ ` − 1,
+ · · · + ` ).
Here we will consider the fundamental K-modules
Λ 1 Cn , Λ 2 Cn , . . . , Λ ` Cn .
Spherical Functions of Fundamental K-Types Associated with the n-Dimensional Sphere
19
We will show that the highest weight of Λp (Cn ) is 1 + · · · + p for 1 ≤ p ≤ `. Also we will
establish that Λp (Cn ) splits into the direct sum of two M -submodules for 1 ≤ p ≤ ` − 1, while
Λ` (Cn ) splits into the sum of three M -submodules; for this reason it will be treated separately
in Section 8.
Observe that λ` is not analytically integral and therefore it will not be considered, although
we will consider the K-module with highest weight 2λ` .
As in the even case we are interested in computing
n−1
X
π̇(Inj )Ps π̇(Inj )
j=1
Vr
= λ(r, s)IVr ,
with r, s = 0, 1 corresponding to the two M -submodules V0 and V1 of the representation π,
corresponding to mn−1 = (1, . . . , 1, 0, . . . , 0) ∈ C` with p − 1 and p ones respectively (see the
betweenness conditions in Section 2.5). Being P0 and P1 the respective projections.
Let us consider the standard action of K = SO(n) on V = Cn , and take the canonical basis
{e1 , . . . , en }. Then we have the irreducible K-module Λp (V ) for 1 ≤ p ≤ ` − 1. The vector
(e1 − ie2 ) ∧ (e3 − ie4 ) ∧ · · · ∧ (e2p−1 − ie2p ) is the unique, up to a scalar, dominant vector and
its weight is (1, . . . , 1, 0, . . . , 0) ∈ C` with p ones. Then, if V 0 is the subspace generated by
{e1 , . . . , en−1 }, Λp (V ) is the direct sum of two irreducible M -submodules, namely
Λp (V ) = V0 ⊕ V1 = Λp−1 (V 0 ) ∧ en ⊕ Λp (V 0 )
(5.2)
of highest weights (1, . . . , 1, 0, . . . , 0) ∈ C` with p − 1 ones, and (1, . . . , 1, 0, . . . , 0) ∈ C` with p
ones, respectively. It is easy to see that (e1 − ie2 ) ∧ (e3 − ie4 ) ∧ · · · ∧ (e2p−3 − ie2p−2 ) ∧ en is
an M -highest weight vector in Λp−1 (V 0 ) ∧ en and that (e1 − ie2 ) ∧ (e3 − ie4 ) ∧ · · · ∧ (e2p−1 − ie2p )
is an M highest weight vector in Λp (V 0 ).
To get λ(0, 0) it is enough to compute
n−1
X
π̇(Inj )P0 π̇(Inj )(e1 ∧ · · · ∧ ep−1 ∧ en ).
j=1
Since we have that π̇(Inj )(e1 ∧ · · · ∧ ep−1 ∧ en ) = e1 ∧ · · · ∧ ep−1 ∧ ej , we obtain P0 π̇(Inj )(e1 ∧
· · · ∧ ep−1 ∧ en ) = 0 and λ(0, 0) = 0.
To get λ(0, 1) it is enough to compute
n−1
X
π̇(Inj )P1 π̇(Inj )(e1 ∧ · · · ∧ ep−1 ∧ en ).
j=1
We have
(
0
P1 π̇(Inj )(e1 ∧ · · · ∧ ep−1 ∧ en ) =
e1 ∧ · · · ∧ ep−1 ∧ ej
if 1 ≤ j ≤ p − 1,
if p ≤ j ≤ n − 1.
Therefore
(
0
π̇(Inj )P1 π̇(Inj )(e1 ∧ · · · ∧ ep−1 ∧ en ) =
−e1 ∧ · · · ∧ ep−1 ∧ en
Hence λ(0, 1) = −(n − p).
Similarly, to get λ(1, 0) it is enough to compute
n−1
X
j=1
π̇(Inj )P0 π̇(Inj )(e1 ∧ · · · ∧ ep ).
if 1 ≤ j ≤ p − 1,
if p ≤ j ≤ n − 1.
20
J.A. Tirao and I.N. Zurrián
We have that
(
−e1 ∧ · · · ∧ en ∧ · · · ∧ ep
π̇(Inj )(e1 ∧ · · · ∧ ep ) =
0
where en appears in the j-place. Therefore
(
−e1 ∧ · · · ∧ ep
π̇(Inj )P0 π̇(Inj )(e1 ∧ · · · ∧ ep ) =
0
if 1 ≤ j ≤ p,
if p + 1 ≤ j ≤ n − 1,
if 1 ≤ j ≤ p,
if p + 1 ≤ j ≤ n − 1.
Hence λ(1, 0) = −p.
Also it is clear now that
n−1
P
π̇(Inj )P1 π̇(Inj )(e1 ∧ · · · ∧ ep ) = 0, hence λ(1, 1) = 0.
j=1
Therefore, when π is the standard representation of K in Λp (V ), 1 ≤ p ≤ ` − 1, we have
0 p−n
(λ(r, s))0≤r,s≤1 =
.
−p
0
Therefore, we obtain a more explicit version of Corollary 3.6 using (3.2) and Remark 3.8.
ˆ
Corollary 5.2. Let Φ be an irreducible spherical function on G of type π ∈ SO(n),
n = 2` + 1.
`
If the highest weight of π is of the form (1, . . . , 1, 0, . . . , 0) ∈ C , with p ones, 1 ≤ p ≤ ` − 1, then
the function H : (0, 1) → End(C2 ) associated with Φ satisfies
1
1 + (1 − 2y)2 p − n 0
00
0
y(1 − y)H (y) + n(1 − 2y)H (y) +
H(y)
0
−p
2
4y(1 − y)
(1 − 2y)
0 p−n
+
H(y) = λH(y),
0
2y(1 − y) −p
for some λ ∈ C.
6
The spherical functions of fundamental K-types
Let n = 2`, the irreducible spherical functions of K-type
mn = (1, . . . , 1, 0, . . . , 0) ∈ C` ,
with p ones, 1 ≤ p ≤ ` − 1, are those associated with the irreducible representations of G of
highest weights of the form mn+1 = (w + 1, 1, . . . , 1, δ, 0, . . . , 0) ∈ C` that interlaces mn ,
w+1
1
1 ...
...
1
...
δ
1
0
0 ...
...
0
...
0
.
We now consider the K-module Λp (Cn ) which has highest weight mn .
For w = 0 and δ = 0 we consider the G-module Λp (Cn+1 ) whose highest weight is mn+1 , and
we have the following K-module decomposition
Λp Cn+1 = Λp (Cn ) ⊕ Λp−1 Cn ∧ en+1 ,
where Λp (Cn ) is the sum of two SO(n − 1)-modules:
Λp Cn = Λp Cn−1 ⊕ Λp−1 Cn−1 ∧ en .
Spherical Functions of Fundamental K-Types Associated with the n-Dimensional Sphere
21
We observe that
a(s)(e1 ∧ · · · ∧ ep−1 ∧ en ) = e1 ∧ · · · ∧ ep−1 ∧ (cos sen − sin sen+1 )
= cos s(e1 ∧ · · · ∧ ep−1 ∧ en ) − sin s(e1 ∧ · · · ∧ ep−1 ∧ en+1 ).
Hence, if Φ0 is the spherical function associated with the irreducible representation of G of
highest weight mn+1 = (1, 1, . . . , 1, δ, 0, . . . , 0) ∈ C` with δ = 0, we have that
Φ0 (a(s))(e1 ∧ · · · ∧ ep−1 ∧ en ) = cos s(e1 ∧ · · · ∧ ep−1 ∧ en ).
Also we have that a(s)(e1 ∧ · · · ∧ ep ) = e1 ∧ · · · ∧ ep . Thus the vector valued function F0 (s) given
by the irreducible spherical function Φ0 (a(s)) is
cos s
F0 (s) =
.
1
For w = 0 and δ = 1 we consider the G-module Λp+1 (Cn+1 ) whose highest weight mn+1 , and
for 1 ≤ p ≤ ` − 1 we have the following K-module decomposition
Λp+1 Cn+1 = Λp+1 Cn ⊕ Λp Cn ∧ en+1 ,
where Λp (Cn ) ∧ en+1 is the sum of two SO(n − 1)-modules:
Λp Cn ∧ en+1 = Λp Cn−1 ∧ en+1 ⊕ Λp−1 Cn−1 ∧ en ∧ en+1 .
We observe that
a(s)(e1 ∧ · · · ∧ ep−1 ∧ en ∧ en+1 ) = e1 ∧ · · · ∧ ep−1 ∧ (sin sen + cos sen+1 )
= sin s(e1 ∧ · · · ∧ ep−1 ∧ en ) + cos s(e1 ∧ · · · ∧ ep−1 ∧ en+1 ).
Hence, if Φ1 is the spherical function associated with the irreducible representation of G of
highest weight mn+1 = (1, 1, . . . , 1, δ, 0, . . . , 0) ∈ C` with δ = 1, we have that Φ1 (a(s))(e1 ∧ · · · ∧
ep−1 ∧ en ∧ en+1 ) = cos s(e1 ∧ · · · ∧ ep−1 ∧ en ∧ en+1 ). Also we have that
a(s)(e1 ∧ · · · ∧ ep ∧ en+1 ) = e1 ∧ · · · ∧ ep−1 ∧ en ∧ en+1 .
Thus the vector valued function F1 (s) given by the irreducible spherical function Φ1 (a(s)) is
1
F1 (s) =
.
cos s
Definition 6.1. We shall consider the 2 × 2 matrix-valued function Ψ = Ψ(y), for 0 < y < 1,
whose columns are given by the functions H0 (y) = F0 (s) and H1 (y) = F1 (s), with cos s = 2y −1:
2y − 1
1
Ψ(y) =
.
(6.1)
1
2y − 1
Since the functions H0 (y) and H1 (y) are associated with irreducible spherical functions, they
satisfy the differential equation given in Corollary 5.1; moreover, the respective eigenvalues are
λ = −p and λ = p − n. Therefore, we have
1
1 + (1 − 2y)2 p − n 0
00
0
y(1 − y)Ψ + n(1 − 2y)Ψ +
Ψ
0
−p
2
4y(1 − y)
(1 − 2y)
0 p−n
−p
0
+
Ψ=Ψ
.
0
0 p−n
2y(1 − y) −p
Furthermore, it is easy to check that the function Ψ(y) also satisfy the equation above even
when n is odd.
22
J.A. Tirao and I.N. Zurrián
Theorem 6.2. The function Ψ can be used to obtain a hypergeometric differential equation
from the one given in Corollaries 5.1 and 5.2. Precisely, if H is a vector-valued solution of the
differential equation in Corollaries 5.1 or 5.2, with eigenvalue λ, then P = Ψ−1 H is a solution
of DP = λP , where D is the hypergeometric differential operator given by
n
( 2 + 1)(2y − 1)
−1
p
0
00
0
DP = y(1 − y)P −
P −
P.
−1
( n2 + 1)(2y − 1)
0 n−p
Proof . By hypothesis we have that
1
1 + (1 − 2y)2
y(1 − y)H (y) + n(1 − 2y)H 0 (y) +
2
4y(1 − y)
(1 − 2y)
0 p−n
+
H(y) = λH(y),
−p
0
2y(1 − y)
00
p−n 0
H(y)
0
−p
Then, writing H = ΨP , we have
n
y(1 − y)P 00 + 2y(1 − y)Ψ−1 Ψ0 + (1 − 2y)I P 0
2
n
1 + (1 − 2y)2 p − n 0
+ Ψ−1 y(1 − y)Ψ00 + (1 − 2y)Ψ0 +
Ψ
0
−p
2
4y(1 − y)
(1 − 2y)
0 p−n
+
Ψ P = λP.
0
2y(1 − y) −p
Now we compute
−1
2y(1 − y)Ψ
4y(1 − y)
Ψ =
4y(y − 1)
0
2y − 1
−1
−1
2y − 1
2y − 1
−1
.
=−
−1
2y − 1
Therefore
n
( 2 + 1)(2y − 1)
−1
λ+p
0
0
y(1 − y)P −
P = 0.
P −
−1
( n2 + 1)(2y − 1)
0
λ+n−p
00
This completes the proof of the theorem.
6.1
∆-eigenvalues of spherical functions
As we said, when n = 2` the irreducible spherical functions of the pair (SO(n+1), SO(n)), of type
mn = (1, . . . , 1, 0 . . . , 0) ∈ C` with p ones, 1 ≤ p ≤ ` − 1 are those associated with the irreducible
representations τ of G of highest weights of the form mn+1 = (w + 1, 1, . . . , 1, δ, 0, . . . , 0) ∈ C`
with p − 1 ones, such that the following pattern holds
w+1
1
1 ...
...
1
...
δ
1
0
0 ...
...
0
...
0
.
Let Φw,δ be the corresponding spherical function. Then ∆Φw,δ = λΦw,δ , where the eigenvalue
λ = λn (w, δ) can be computed from the expression ∆ = Qn+1 − Qn . If v ∈ Vmn+1 is a highest
weight vector from (2.6) we have
τ̇ (Q2`+1 )v = − (w + 1)2 + (2` − 1)(w + 1) + (2` − p)(p − 1) + 2δ(` − p) v.
If v ∈ Vm2` is a highest weight vector, then from (2.5) we have
π̇(Qn )v = −p(2` − p)v.
Spherical Functions of Fundamental K-Types Associated with the n-Dimensional Sphere
23
Since ∆ = Qn+1 − Qn it follows that
λ2` (w, δ) = −(w + 1)2 − (2` − 1)(w + 1) + (2` − p) − 2δ(` − p)
Analogously, we obtain that the eigenvalues of the spherical functions Φw,δ of the pair (SO(2` +
2), SO(2` + 1)) are of the form
λ2`+1 (w, δ) = −(w + 1)(w + 2` + 1) + 2` − p + 1 − δ2(` − p) − δ 2 ,
here δ is 0 or 1 when we are in the cases 1 ≤ p < ` but δ could also be −1 in the particular case
p = `.
Therefore, we have that the eigenvalues of the spherical functions Φw,δ of the pair (SO(n +
1), SO(n)) are of the form
(
−w(w + n + 1) − p
if δ = 0,
λn (w, δ) =
(6.2)
−w(w + n + 1) − n + p if δ = ±1.
6.2
Polynomial eigenfunctions of the hypergeometric operator D
Let D be the differential operator on the real line introduced in Theorem 6.2:
DP = y(1 − y)P 00 + (C − yU )P 0 − V P,
(6.3)
with
(n/2 + 1)
1
,
C=
1
(n/2 + 1)
U = (n + 2)I,
p
0
,
V =
0 n−p
where n is of the form 2` or 2` + 1 for ` ∈ N and 1 ≤ p < `.
We will study the C2 -vector valued polynomial eigenfunctions of D.
The equation DP = λP is an instance of a matrix hypergeometric differential equation
studied in [22]. Since the eigenvalues of C, n/2 and n/2 + 2, are not in −N0 the function P is
determined by P0 = P (0). For |y| < 1 it is given by
∞
X
yj
U, V + λ
; y P0 =
P (y) = 2 H1
[C; U ; V + λ]j P0 ,
P0 ∈ C2 ,
C
j!
j=0
where the symbol [C; U ; V + λ]j is inductively defined by
[C; U ; V + λ]0 = 1,
[C; U ; V + λ]j+1 = (C + j)−1 (j(U + j − 1) + V + λ)[C; U ; V + λ]j ,
for all j ≥ 0.
Therefore, we have that there exists a polynomial solution if and only if the coefficient
[C; U ; V + λ]j+1 is a singular matrix for some j ∈ Z. Since the matrix C + j is invertible for all
j ∈ N0 , we have that there is a polynomial solution of degree j for DP = λP if and only if there
exists P0 ∈ C2 such that [C; U ; V + λ]j P0 6= 0 and (j(U + j − 1) + V + λ)[C; U ; V + λ]j P0 = 0.
Now we easily observe that the only possible values for λ such that j(U + j − 1) + V + λ has
non trivial kernel are those given in (6.2). Then, if λ = −w(w + n + 1) − p, it is easy to check
that the first and only j for which j(U + j − 1) + V + λ is singular is j = w, and its kernel (of
dimension 1) is the subspace generated by ( 10 ). Analogously, if λ = −w(w + n + 1) − n + p, it is
easy to check that the first and only j for which j(U + j − 1) + V + λ is singular is j = w, and
its kernel (of dimension 1) is the subspace generated by ( 01 ) respectively. Therefore we have the
following result.
24
J.A. Tirao and I.N. Zurrián
Theorem 6.3. For a given ` ∈ N take n = 2` or 2` + 1 and 1 ≤ p ≤ ` − 1, then the polynomial
eigenfunctions of
DP = y(1 − y)P 00 + (C − yU )P 0 − V P,
with
C=
(n/2 + 1)
1
,
1
(n/2 + 1)
U = (n + 2)I,
V =
p
0
0 n−p
have eigenvalues −w(w + n + 1) − p or −w(w + n + 1) − n + p, with w ∈ N0 ; in both cases the
degree of the polynomial is w with leading coefficient a multiple of ( 10 ) or ( 01 ), respectively.
7
The inner product
Given a finite dimensional irreducible representation π of K in the vector space Vπ let (C(G) ⊗
End(Vπ ))K×K be the space of all continuous functions Φ : G −→ End(Vπ ) such that Φ(k1 gk2 ) =
π(k1 )Φ(g)π(k2 ) for all g ∈ G, k1 , k2 ∈ K. Let us equip Vπ with an inner product such that
π(k) becomes unitary for all k ∈ K. Then we introduce an inner product in the vector space
(C(G) ⊗ End(Vπ ))K×K by defining
Z
hΦ1 , Φ2 i =
tr(Φ1 (g)Φ2 (g)∗ )dg,
G
R
where dg denote the Haar measure on G normalized by G dg = 1, and where Φ2 (g)∗ denotes
the adjoint of Φ2 (g) with respect to the inner product in Vπ .
By using Schur’s orthogonality relations for the unitary irreducible representations of G,
it follows that if Φ1 and Φ2 are non equivalent irreducible spherical functions, then they are
orthogonal with respect to the inner product h·, ·i, i.e.
hΦ1 , Φ2 i = 0.
Recall that, given an irreducible spherical function Φ of type π of the pair (G, K), the function
Φ(a(s)) is scalar valued when restricted to any SO(n − 1)-module (see (3.1) for a(s)). We shall
denote by m the number of SO(n − 1)-submodules of π, and by d1 , d2 , . . . , dm the respective
dimensions of each one of those submodules.
In particular, if Φ1 and Φ2 are two irreducible spherical functions of type π ∈ K̂, we consider
the vector valued functions H1 (y) and H2 (y) given by the diagonal matrix valued functions
Φ1 (a(s)) and Φ2 (a(s)) (see Remark 3.7), with y = (cos s + 1)/2, respectively, denoting
H1 (y) = (h1 (y), . . . , hm (y))t ,
H2 (y) = (f1 (y), . . . , fm (y))t .
Proposition 7.1. If Φ1 , Φ2 are two irreducible spherical functions of type π ∈ K̂ then
Z 1
m
(n − 1)!! 2 X
hΦ1 , Φ2 i =
di
(y(1 − y))n/2−1 hi (y)fi (y)dy,
(n − 2)!! ω∗
0
i=1
with ω∗ = π if n is even and ω∗ = 2 if n is odd.
Proof . Let A = exp RIn+1,n be the Lie subgroup of G of all elements of the form


In−1
0
0
cos s sin s  ,
a(s) = exp sIn+1,n =  0
s ∈ R,
0
− sin s cos s
where In−1 denotes the identity matrix of size n − 1.
Spherical Functions of Fundamental K-Types Associated with the n-Dimensional Sphere
25
Now [12, Theorem 5.10, p. 190] establishes that for every f ∈ C(G/K) and a suitable constant c∗
Z π
Z
Z
δ∗ (a(s))f (ka(s)K)ds dkM ,
f (gK)dgK = c∗
K/M
G/K
−π
where
dgK and
R
R dkM are respectively the invariant measures on G/K and K/M normalized by
G/K dgK = K/M dkM = 1 and the function δ∗ : A −→ R is defined by
δ∗ (a(s)) =
Y
| sin isν(In+1,n )|,
ν∈Σ+
with Σ+ the set of those positive roots whose restrictions to a, the Lie algebra of A, are not
zero. In our case we have δ∗ (a(s)) = | sinn−1 s|.
To find the value of c∗ we consider the function f ≡ 1, having then
Z π
sinn−1 sds.
1 = 2c∗
0
Since
Z
n−1
sin
1
n−2
sinn−2 s cos s +
sds = −
n−1
n−1
Z
sinn−3 ds,
we obtain that, for n = 2` or 2` + 1,
Z π
Z
n−2n−4
n − 2` + 1 π n−2`
sinn−1 sds =
···
sin
sds.
n−1n−3
n − 2` + 2 0
0
Therefore
c∗ =
(n − 1)!! 1
,
(n − 2)!! 2ω∗
with ω∗ = π for n = 2` and ω∗ = 2 for 2` + 1.
Since the function g 7→ tr(Φ1 (g)Φ2 (g)∗ ) is invariant under left and right multiplication by
elements in K, we have
Z
Z π
∗
hΦ1 , Φ2 i =
tr(Φ1 (g)Φ2 (g) )dg = 2c∗
sinn−1 s tr (Φ1 (a(s)Φ2 (a(s))∗ )) ds.
G
0
If we put y = 21 (cos s + 1) for 0 < s < π we have
∗
tr (Φ1 (a(s)Φ2 (a(s)) )) =
m
X
di hi (y)fi (y).
i=1
Then
hΦ1 , Φ2 i = 4c∗
m
X
i=1
Z
1
di
(4y(1 − y))(n−2)/2 hi (y)fi (y)dy,
0
and the proposition follows.
Proposition 7.2. If Φ1 , Φ2 ∈ (C ∞ (G) ⊗ End(Vπ ))K×K then
h∆Φ1 , Φ2 i = hΦ1 , ∆Φ2 i.
26
J.A. Tirao and I.N. Zurrián
Proof . If we apply a left invariant vector field X ∈ g, to the function on G given by g 7→
tr(Φ1 (g)Φ2 (g)∗ ), and then we integrate over G we obtain
Z
Z
∗
tr (Φ1 (g)(XΦ2 )(g)∗ ) dg.
tr ((XΦ1 )(g)Φ2 (g) ) dg +
0=
G
G
Therefore hXΦ1 , Φ2 i = −hΦ1 , XΦ2 i. Now let τ : gC −→ gC be the conjugation of gC with
respect to the real linear form g. Then −τ extends to a unique antilinear involutive ∗ operator
on D(G) such that (D1 D2 )∗ = D2∗ D1∗ for all D1 , D2 ∈ D(G). This follows easily from the fact
that the universal enveloping algebra over C of g is canonically isomorphic to D(G). Then it
follows that hDΦ1 , Φ2 i = hΦ1 , D∗ Φ2 i.
Finally, it is easy to verify that ∆∗ = ∆.
7.1
Spherical functions as polynomial solutions of DP = λP
e the differential operator on (0, 1) introduced in Corollaries 5.1 and 5.2:
Let us consider D,
1
1 + (1 − 2y)2 p − n 0
00
0
y(1 − y)H (y) + n(1 − 2y)H (y) +
H(y)
0
−p
2
4y(1 − y)
(1 − 2y)
0 p−n
H(y) = λH(y),
(7.1)
+
0
2y(1 − y) −p
e −1
Recall that the operator D that appears in (6.3) extends the differential operator D = ΨDΨ
to the whole real line, where
2y − 1
1
Ψ(y) =
1
2y − 1
is the matrix function given in (6.1) and used in Theorem 6.2.
We want to focus our attention on the following vector spaces of C2 -valued analytic functions
on (0, 1):
e = λH, H( cos s+1 ) analytic at s = 0 ,
Sλ = H = H(y) : DH
2
Wλ = P = P (y) : DP = λP, analytic on [0, 1] .
From Theorem 6.2 we know that the correspondence P 7→ ΨP is an injective linear map
from Wλ into Sλ . Now we want to prove that this map is bijective.
Theorem 7.3. The linear map P 7→ ΨP is an isomorphism from Wλ onto Sλ .
Proof . A vector valued function P ∈ Wλ is an eigenfunction of the hypergeometric operator D.
Since it is analytic at y = 1 it is determined by P (1), therefore dim(Wλ ) = 2.
On the other hand, if H ∈ Sλ then there is a function F (s) analytic at s = 0, such that it
extends the function H( cos2s+1 ) defined on (0, π). Then, F satisfies the following differential
equation
cos s 0
1 + cos2 s p − n 0
00
F (s) + (n − 1)
F (s) +
F (s)
0
−p
sin s
sin2 s
cos s
0 p−n
−2 2
F (s) = λF (s),
0
sin s −p
or equivalently
n−1
p−n 0
0
2
sin sF (s) +
sin(2s)F (s) + (2 − sin s)
F (s)
0
−p
2
2
00
Spherical Functions of Fundamental K-Types Associated with the n-Dimensional Sphere
0 p−n
− 2 cos s
F (s) = λ sin2 sF (s),
−p
0
27
(7.2)
Let aj ∈ C2 and αj , βj , γj ∈ C, for j ≥ 0, be the Taylor coefficients of F , sin, sin2 and cos at
s = 0:
X
X
αj sj ,
aj sj ,
sin s =
F (s) =
j≥1
j≥0
F 0 (s) =
X
aj+1 (j + 1)sj ,
sin2 s =
F (s) =
X
βj sj ,
j≥2
j≥0
00
X
j
aj+2 (j + 2)(j + 1)s ,
cos s =
X
γj sj .
j≥0
j≥0
Therefore, from (7.2) we have
" j−2
j−1
X X
n − 1 X j−k
p−n 0
βj−k ak+2 (k + 2)(k + 1) +
2 αj−k ak+1 (k + 1) +
0
−p
2
j≥0 k=0
k=0
#
#
" j−2
!
j
j−2
X X
X
0 p−n X
γj−k ak sj = λ
βj−k ak sj .
βj−k ak − 2
× 2aj −
−p
0
k=0
k=0
j≥0
k=0
Hence, since β2 = α1 = γ0 = 1, we have that
p − n −p + n
aj
j(j − 1) + (n − 1)j + 2
p
−p
is a linear combination with matrix coefficients of {a0 , a1 , . . . , aj−1 }; it is clear that for j = 1 and
j > 2 the matrix above is non singular, therefore {a0 , a2 } determine completely the sequence
{aj }j≥0 . Also it is clear that when j = 0 or 2, that matrix has nullity 1. Therefore we can
conclude that dim(Sλ ) = 2. The theorem follows.
Theorem 7.4. Let H be the C2 -valued analytic function on (0, 1) given by an irreducible spherical function Φ on G of fundamental K-type (1, . . . , 1, 0, . . . , 0) ∈ C` , with p ones, 0 < p < `.
If P = Ψ−1 H, then P is polynomial.
Proof . We know that the function H is analytic in (0, 1), and from Corollary 5.1 we know that
e (see (7.1)). Also we know that the function H( 1+cos s )
it is an eigenfunction of the operator D
2
is analytic at s = 0, since Φ(a(s)) it is. Therefore from Theorem 7.3 the function P = Ψ−1 H is
an analytic eigenfunction of D on the closed interval [0, 1].
If we introduce the following matrix-weight function V = V (y) supported on the interval [0, 1]
(n − 1)!! 2
0
n/2−1 d1
V (y) =
(y(1 − y))
,
0 d2
(n − 2)!! ω∗
with ω∗ = π if n is even and ω∗ = 2 if n is odd, then from Proposition 7.1 we have
Z 1
hΦ0 , Φ1 i =
H2∗ (y)V (y)H1∗ (y)dy.
0
e is a symmetric operator with respect to the
It follows from Propositions 7.1 and 7.2 that D
inner product defined among continuous vector-valued functions on [0, 1] by
Z 1
hH1 , H2 iV =
H2∗ (y)V (y)H1 (y)dy.
0
28
J.A. Tirao and I.N. Zurrián
e
Then, since D = Ψ−1 DΨ,
we have that D is a symmetric operator with respect to the inner
product defined among continuous vector-valued functions on [0, 1] by
Z 1
hP1 , P2 iW =
P2∗ (y)W (y)P1 (y)dy,
0
where
W = Ψ∗ V Ψ.
Actually, we have that (W, D) is a classical pair in the sense of [7], see also [2]. As the weight W
has finite moments there exists a sequence {Qr }r≥0 of 2 × 2 matrix-valued orthonormal polynomials, such that DQr = Qr Λr where Λr is a real diagonal matrix (for precise definitions and
general facts on matrix-valued orthogonal polynomials see [5] and [2]).
Let {e1 , e2 } be the canonical basis of C2 . Then
Z 1
∗
∗
∗
hQr ej , Qs ei iW = ei
Qs (y)W (y)Qr (y)dy ej = e∗i δsi Iej = δr,s δi,j .
0
Therefore, for r ≥ 0, j = 1, 2, {Qr ej } is a family of C2 -valued orthonormal polynomials such
that
D(Qr ej ) = (DQr )ej = (Qr Λr )ej = Qr (Λr ej ) = λjr (Qr ej ),
where Λr = diag(λ1r , λ2r ).
P
ar,j Qr ej , where ar,j = hP, Qr ej iW . Since P
Now we write our function P = Ψ−1 H as P =
r,j
is analytic on [0, 1] the sum converges not only in the L2 -norm but also in the topology based
on uniform convergence of sequences of functions and their successive derivatives.
Therefore,
X
λP = DP =
ar,j λjr Qr ej .
r,j
Then ar,j = 0 if λjr 6= λ. Since dim Wλ = 2 it follows that P is a polynomial.
Remark 7.5. It is easy to see from (5.1) and (5.2) that the dimensions of the M -submodules of
the fundamental representation of K with highest weight of the form (1, . . . , 1, 0 . . . , 0), with p
ones, are given by
d1 =
(n − 1)!
,
(p − 1)!(n − p)!
d2 =
(n − 1)!
,
p!(n − 1 − p)!
therefore the weight W is given by
W =
(n − 1)!! 2 (n − 1)!
p
0
(y(1 − y))n/2−1 Ψ∗
Ψ,
0 n−p
(n − 2)!! ω∗ p!(n − p)!
with ω∗ = π if n is even and ω∗ = 2 if n is odd. Then, W is a scalar multiple of
p(2y − 1)2 + n − p
n(2y − 1)
.
n(2y − 1)
(n − p)(2y − 1)2 + p
Even more, since 0 < p < ` and n = 2`, 2` + 1 it follows that p 6= n − p. Then it can be proved
that the weight W does not reduce to a smaller size, i.e., there is not any invertible matrix M
such that M ∗ W (y)M is diagonal for all y ∈ [0, 1].
Spherical Functions of Fundamental K-Types Associated with the n-Dimensional Sphere
29
ˆ
For a given fundamental K type π ∈ SO(n),
n = 2` or 2` + 1, with highest weight of the
`
form (1, . . . , 1, 0, . . . , 0) ∈ C with p ones (0 < p < `), let Φw,δ denote the irreducible spherical
ˆ
function of the pair (SO(n + 1), SO(n)) given by τ ∈ SO(n
+ 1) with highest weight of the form
(w + 1, 1, . . . , 1, δ, 0 . . . , 0) with p − 1 ones.
Therefore, combining (6.2), Theorems 6.3 and 7.4 we have the following statement.
Theorem 7.6. Given w ∈ N0 , every irreducible spherical function Φw,δ of the pair (SO(n + 1),
SO(n)), with n = 2` or 2` + 1, of type mn = (1, . . . , 1, 0, . . . , 0) ∈ C` with p ones (0 < p < `),
corresponds to a vector valued function Pw,δ (δ = 0, 1), which is a polynomial of degree w; and
the leading coefficients of Pw,0 and Pw,1 are multiples of ( 10 ) and ( 01 ) respectively. Precisely
Pw,δ (y) =
w
X
yj
j=0
j!
[C; U ; V + λ]j Pw,δ (0),
with
p
0
(n/2 + 1)
1
,
,
U = (n + 2)I,
V =
C=
0 n−p
1
(n/2 + 1)
(
−w(w + n + 1) − p
if δ = 0,
λ = λn (w, δ) =
−w(w + n + 1) − n + p if δ = 1.
Even more, the value of Pw,δ (0) can be computed.
Proof . It only remains to prove that Pw,δ (0) can be computed.
Let us consider the case δ = 0. We know from (6.2) and Theorem 6.3 that there is some
c ∈ C such that
[C; U ; V + λ]w Pw,0 (0) = c
1
.
0
Since [C; U ; V + λ]w is invertible, this c is univocally determined by the condition Φ(e) = I,
which implies
w
X
1
1
Ψ(1)
[C; U ; V + λ]j Pw,0 (0) =
.
1
j!
j=0
Similarly, we can prove the same for Pw,1 (0).
Remark 7.7. It is worth to observe that for w, w0 ≥ 0 and δ, δ 0 = 0, 1, since hPw,δ , Pw0 ,δ0 iW =
hΦw,δ , Φw0 ,δ0 i, we have that if (w, δ) 6= (w0 , δ 0 ) then
hPw,δ , Pw0 ,δ0 iW = 0.
Therefore, our construction encodes all equivalent classes of irreducible spherical functions of
a fundamental K-type of highest weight λp , 0 < p < `, in the orthogonal set of C2 -valued
polynomials {Pw,0 , Pw,1 }. The degree of Pw,0 and Pw,1 is w, and the leading coefficient is
a multiple of ( 10 ) or ( 01 ), respectively.
30
8
J.A. Tirao and I.N. Zurrián
Matrix valued orthogonal polynomials
8.1
Matrix valued orthogonal polynomials
In this subsection, given n of the form 2` or 2` + 1 with ` ∈ N, for a fixed 0 < p < ` we
shall construct a sequence of matrix-valued polynomials {Pw }w≥0 directly related to irreducible
ˆ
spherical functions of type π ∈ SO(n)
of highest weight mπ = (1, . . . , 1, 0 . . . , 0) ∈ C` , with p
ones.
Given a nonnegative integer w and δ = 0, 1, we can consider Φw,δ , the irreducible spherical
ˆ
function of type π associated with the irreducible representation τ ∈ SO(n
+ 1) of highest weight
of the form mτ = (w + 1, 1, . . . , 1, δ, 0, . . . , 0) with p − 1 ones.
We insist on recalling that, since π has only two SO(n − 1)-submodules, we can interpret the
diagonal matrix-valued function Φw,δ (a(s)), s ∈ (0, π), as a 2 column vector function.
Now we consider the vector-valued function
Pw,δ : (0, 1) → C2
given by the vector function Pw,δ (y) = Ψ−1 (y)Φw,δ (a(s)), with cos(s) = 2y − 1. Then, we define
the matrix-valued function
Pw = Pw (y),
whose δ-th column (δ = 0, 1) is given by the C2 -valued polynomial Pw,δ (y).
Let consider the matrix-valued skew symmetric bilinear form defined among C ∞ 2×2 matrixvalued functions on [0, 1] by
Z 1
hP, QiW =
Q∗ (y)W (y)P (y)dy,
0
where
2
(n − 1)!! 2 (n − 1)!
n(2y − 1)
n/2−1 p(2y − 1) + n − p
W =
.
(y(1 − y))
n(2y − 1)
(n − p)(2y − 1)2 + p
(n − 2)!! ω∗ p!(n − p)!
See Remark 7.5. Then we state the following theorem.
Theorem 8.1. The matrix-valued polynomial functions Pw , w ≥ 0, form a sequence of orthogonal polynomials with respect to W , which are eigenfunctions of the symmetric differential
operator D in (6.3). Moreover,
λ(w, 0)
0
DPw = Pw
,
0
λ(w, 1)
where
(
−w(w + n + 1) − p
λ(w, δ) =
−w(w + n + 1) − n + p
if
if
δ = 0,
δ = 1.
Proof . From Theorem 6.2 we have that the δ-th column of Pw is an eigenfunction of the
operator D with eigenvalue λ(w, δ), see (6.2) and (6.3). Therefore we have
DPw = Pw Λw ,
with
Λw =
λ(w, 0)
0
.
0
λ(w, 1)
Spherical Functions of Fundamental K-Types Associated with the n-Dimensional Sphere
31
From Theorem 7.6 we know that each column of Pw is a polynomial function of degree w and,
even more, that Pw is a polynomial whose leading coefficient is a nonsingular diagonal matrix.
Given w and w0 , non negative integers, by using Remark 7.7 we have
Z
hPw0 , Pw iW =
1
∗
Pw (y) W (y)Pw0 (y)du =
0
=
1
X
1
Pw,δ (y)∗ W (y)Pw0 ,δ0 (y)du Eδ,δ0
δ,δ 0 =0 0
Z
δw,w0 δδ,δ0
1
Pw,δ (y) W (y)Pw0 ,δ0 (y)du Eδ,δ0
∗
0
δ,δ 0 =0
= δw,w0
1 Z
X
1 Z
X
δ=0
1
Pw,δ (y)∗ W (y)Pw0 δ (y)du, Eδ,δ ,
0
which proves the orthogonality. Even more, it also shows us that hPw , Pw iW is a diagonal matrix.
Also, making a few simple computations we have that
hDPw , Pw0 i = δw,w0 hPw , Pw0 iΛw = δw,w0 Λ∗w hPw , Pw0 i = hPw , DPw0 i,
for every w, w0 ∈ N0 , since Λw is real and diagonal. This concludes the proof of the theorem.
9
The SO(2` + 1)-type with highest weight 2λ`
In this section K = SO(2` + 1). We will focus on the particular case when the K-type is given
by an irreducible representation π with highest weight 2λ` = (1, 1, . . . , 1). We will first see that
such K-module is the direct sum of three M -submodules, and we will find similar results to those
obtained for the fundamental K-types λ1 , . . . , λ`−1 that are direct sum of two M -submodules.
Let us consider the irreducible K-module Λ` (V ), with V = Cn , n = 2` + 1. The vector
v = (e1 − ie2 ) ∧ (e3 − ie4 ) ∧ · · · ∧ (e2`−1 − ie2` ) is the unique, up to a scalar, dominant vector
and its weight is 2λ` = (1, 1, . . . , 1).
It is not difficult to see that Λ` (V ) is the sum of three M -irreducible submodules, namely
Λ` (V ) = V1 ⊕ V0 ⊕ V−1
(9.1)
with respective highest weights (1, . . . , 1), (1, . . . , 1, 0), (1, . . . , 1, −1) ∈ C` and having V0 =
Λ`−1 (V ) ∧ en and V1 ⊕ V−1 ' Λ` (Cn−1 ).
The vectors
v1 = (e1 − ie2 ) ∧ (e3 − ie4 ) ∧ · · · ∧ (e2`−1 − ie2` ),
v0 = −(e1 − ie2 ) ∧ (e3 − ie4 ) ∧ · · · ∧ (e2`−3 − ie2`−2 ) ∧ en ,
v−1 = (e1 − ie2 ) ∧ (e3 − ie4 ) ∧ · · · ∧ (e2`−1 + ie2` )
are M -highest weight vectors in V1 , V0 and V−1 , respectively. Also let us call P1 , P0 and P−1
the respective projections on V1 , V0 and V−1 , according to the decomposition (9.1).
In order to obtain the explicit expression of E in (3.2) we are interested to compute
n−1
X
π̇(Inj )Ps π̇(Inj )
Vr
= λ(r, s)IVr ,
j=1
with r, s = 1, 0, −1 corresponding to the three M -submodules V1 , V0 and V−1 of the representation π.
32
J.A. Tirao and I.N. Zurrián
If 1 ≤ j ≤ `, then
(
0
π̇(In,2j−1 )(e2k−1 − ie2k ) =
−en
(
0
if
π̇(In,2j )(e2k−1 − ie2k ) =
ien if
if k 6= j,
if k = j,
k=
6 j,
k = j,
therefore, it is easy to see that P0 π̇(In,2j−1 )v0 = P0 π̇(In,2j )v0 = 0 and that Pr π̇(In,2j−1 )vs =
Pr π̇(In,2j )vs = 0 when s ± 1 and r ± 1; i.e.
λ(0, 0) = λ(−1, −1) = λ(1, −1) = λ(−1, 1) = λ(1, 1) = 0.
Furthermore, it is easy to see that, for 1 ≤ j ≤ ` and r equal to 1 or −1, we have
π̇(In,2j−1 )P0 π̇(In,2j−1 )vr + π̇(In,2j )P0 π̇(In,2j )vr = −vr ,
then λ(−1, 0) = λ(1, 0) = −`. Therefore, it only remains to compute
`
X
(π̇(In,2j−1 )Ps π̇(In,2j−1 )v0 + π̇(In,2j )Ps π̇(In,2j )v0 ) ,
j=1
for s = ±1.
To obtain Ps π̇(In,k )v0 it is necessary to decompose π̇(In,k )v0 according to the direct sum (9.1).
We know that π̇(X−εj −ε` )v1 ∈ V1 and π̇(X−εj +ε` )v−1 ∈ V−1 ; recall that
X−j −` = I2`−1,2j−1 − I2`,2j + i(I2`−1,2j + I2`,2j−1 ),
X−j +` = I2`−1,2j−1 + I2`,2j + i(I2`−1,2j − I2`,2j−1 ),
see (2.2). We have
π̇ X−εj −ε` (e2j−1 − ie2j ) = −2(e2`−1 + ie2` ),
π̇ X−εj −ε` (e2`−1 − ie2` ) = 2 (e2j−1 + ie2j ) ,
π̇ X−εj −ε` (e2k−1 − ie2k ) = 0,
for k 6= s, `.
Therefore, for 1 ≤ j < `,
π̇ X−εj −ε` v1 = 2(e1 − ie2 ) ∧ · · · ∧ (e2(`−1)−1 − ie2(`−1) ) ∧ (e2j−1 + ie2j )
− 2(e1 − ie2 ) ∧ · · · ∧ (e2j−3 − ie2j−2 ) ∧ (e2`−1 + ie2` ) ∧ (e2j+1 − ie2j+2 )∧
· · · ∧ (e2`−1 − ie2` ).
Similarly, for 1 ≤ j < `,
π̇ X−εj +ε` v1 = 2(e1 − ie2 ) ∧ · · · ∧ (e2(`−1)−1 − ie2(`−1) ) ∧ (e2j−1 + ie2j )
+ 2(e1 − ie2 ) ∧ · · · ∧ (e2j−3 − ie2j−2 ) ∧ (e2`−1 + ie2` ) ∧ (e2j+1 − ie2j+2 )∧
· · · ∧ (e2`−1 − ie2` ).
Hence, for 1 ≤ j < `, we have
−i
8 π̇ X−εj −ε` v1 + π̇ X−εj +ε` v−1
1
8
= (e1 − ie2 ) ∧ · · · ∧ (e2(`−1)−1 − ie2(`−1) ) ∧ e2j = π̇(In,2j )v0 ,
π̇ X−εj −ε` v1 + π̇ X−εj +ε` v−1
Spherical Functions of Fundamental K-Types Associated with the n-Dimensional Sphere
33
= (e1 − ie2 ) ∧ · · · ∧ (e2(`−1)−1 − ie2(`−1) ) ∧ e2j−1 = π̇(In,2j−1 )v0 .
Then, for 1 ≤ j < `,
π̇ (In,2j−1 ) P1 π̇ (In,2j−1 ) v0 = 18 π̇ (In,2j−1 ) π̇ X−εj −ε` v1
i
2
e1 − ie2 ) ∧ · · · ∧ e2j ∧ · · · ∧ (e2(`−1)−1 − ie2(`−1) ) ∧ en ,
π̇ (In,2j ) P1 π̇ (In,2j ) v0 = −i
8 π̇ (In,2j ) π̇ X−εj −ε` v1
=
= − 12 e1 − ie2 ) ∧ · · · ∧ e2j−1 ∧ · · · ∧ (e2(`−1)−1 − ie2(`−1) ) ∧ en .
Therefore, for 1 ≤ j < `,
π̇ (In,2j−1 ) P1 π̇ (In,2j−1 ) v0 + π̇ (In,2j ) v0 P1 π̇ (In,2j ) v0 = − 21 v0 .
Besides, for j = ` we have
π̇(In,2` )v0 =
1
2i (−v1
+ v−1 )
and
π̇(In,2`−1 )v0 = 21 (v1 + v−1 ).
Therefore, since
1
π̇ (In,2` ) v1 = − 21 v0 ,
π̇ (In,2` ) P1 π̇(In,2` )v0 = − 2i
π̇ (In,2`−1 ) P1 π̇(In,2`−1 )v0 = 12 π̇ (In,2`−1 ) v1 = − 12 v0 ,
we have that
n−1
X
π̇ (In,j ) P1 π̇(In,j )v0 = −
j=0
`+1
v0 ,
2
i.e.
λ(0, 1) = −
`+1
.
2
Analogously we obtain
λ(0, −1) = −
`+1
.
2
Hence

0
(λ(r, s))−1≤r,s≤1 = − `+1
2
0

−`
0
.
0 − `+1
2
−`
0
Therefore, we obtain a more explicit version of Corollary 3.6 using (3.2) and Remark 3.8.
Confront Corollary 5.2.
ˆ
Corollary 9.1. Let Φ be an irreducible spherical function on G of type π ∈ SO(n),
n = 2` + 1.
`
If the highest weight of π is of the form (1, . . . , 1) ∈ C , then the function H : (0, 1) → C3
e = λH, for some λ ∈ C with
associated with Φ satisfies DH


−`
0
0
2
e = y(1 − y)H 00 (y) + 1 n(1 − 2y)H 0 (y) + (1 − 2y) + 1  0 −` − 1 0  H(y)
DH
2
4y(1 − y)
0
0
−`


0
−`
0
(1 − 2y)  `+1
`+1 
+
H(y).
− 2
0 − 2
2y(1 − y)
0
−`
0
34
9.1
J.A. Tirao and I.N. Zurrián
Spherical functions of SO(2` + 1)-type 2λ`
Let n = 2` + 1, we now focus on the spherical functions Φw,δ of type mn = (1, . . . , 1) ∈ C` ,
which are associated with the irreducible representations of SO(n + 1) of highest weights of the
form mn+1 = (w + 1, 1, . . . , 1, δ) ∈ C`+1 such that the following pattern holds
w+1
1
1 ...
...
1
...
δ
1
−1
.
As before we make the function Ψ whose columns are given by the spherical functions Φ0,δ ,
δ = −1, 0, 1. When w = 0, this is calculable using [24, p. 364, equation (8)] or alternatively by
considering the G-modules Λ`+1 (Cn+1 ) = V1 ⊕ V−1 and Λ` (Cn+1 ) = V0 and working in the same
way that we already did in the beginning of Section 6 for the 2×2 cases (here Vt , for t = 1, 0, −1,
are the irreducible G-modules with highest weights (1, . . . , 1, t) ∈ C`+1 ).
Therefore, if cos s = 2y − 1 we have
 is

e
1
e−is
1 is
−is )
1 
Ψ(y) =  1
2 (e + e
−is
e
1
eis
p
p


1
2y − 1 − 2i y − y 2
2y − 1 + 2i y − y 2
.
1p
2y − 1
1p
=
2
2
2y − 1 − 2i y − y
1
2y − 1 + 2i y − y
Each column of Ψ satisfies the differential equation given in Corollary 9.1. And it is easy to
check that we have


−`
0
0
2+1
1
(1
−
2y)
 0 −` − 1 0  Ψ(y)
y(1 − y)H 00 (y) + n(1 − 2y)H 0 (y) +
2
4y(1 − y)
0
0
−`




0
−`
0
−` − 1 0
0
(1 − 2y)  `+1
 Ψ(y) = Ψ(y)  0
−`
0 .
+
− 2
0 − `+1
2
2y(1 − y)
0
0 −` − 1
0
−`
0
Theorem 9.2. The function Ψ can be used to obtain a hypergeometric differential equation
from the one given in Corollary 9.1. Precisely, if H is a vector-valued solution of the differential
equation in Corollary 9.1, with eigenvalue λ, then P = Ψ−1 H is a solution of DP = λP , where D
is the hypergeometric differential operator given by
DP = y(1 − y)P 00 + (C − yU )P 0 − V P,
with


(n + 2)/2
1/2
0
,
1
(n + 2)/2
1
C=
0
1/2
(n + 2)/2


−` − 1 0
0

0
−`
0 .
V =
0
0 −` − 1
U = (n + 2)I,
Proof . Let us write H = ΨP . Then
n
y(1 − y)P 00 + 2y(1 − y)Ψ−1 Ψ0 + (1 − 2y)I P 0
2
Spherical Functions of Fundamental K-Types Associated with the n-Dimensional Sphere

2y)2
n
1 + (1 −
(1 − 2y)Ψ0 +
2
4y(1 − y)
 
−`
0
 Ψ P = λP.
0 − `+1
2
−`
0
+ Ψ−1 y(1 − y)Ψ00 +

0
(1 − 2y)  `+1
+
− 2
2y(1 − y)
0
35


−`
0
0
 0 −` − 1 0  Ψ
0
0
−`
Now we compute


(n + 2)/2
1/2
0
n
.
1
(n + 2)/2
1
2y(1 − y)Ψ−1 Ψ0 + (1 − 2y)I = −(n + 2)yI + 
2
0
1/2
(n + 2)/2
Therefore



(n + 2)/2
1/2
0
 P 0
1
(n + 2)/2
1
y(1 − y)P 00 + −(n + 2)yI + 
0
1/2
(n + 2)/2



−` − 1 0
0
−`
0  − λI  P = 0.
+  0
0
0 −` − 1
This completes the proof of the theorem.
We obtain a similar result to Theorem 6.3, with an analogous proof:
Theorem 9.3. For a given ` ∈ N let n = 2` + 1, then the nonzero polynomial eigenfunctions of
DP = y(1 − y)P 00 + (C − yU )P 0 − V P,
with


(n + 2)/2
1/2
0
,
1
(n + 2)/2
1
C=
0
1/2
(n + 2)/2


−` − 1 0
0
−`
0 ,
V = 0
0
0 −` − 1
U = (n + 2)I,
have eigenvalues −w(w + n + 1) − ` or −w(w + n + 1) − ` − 1, with w ∈ N0 . In both
cases
0
the degree of the polynomial is w and the leading coefficient can be any multiple of 1 or any
0
0
1
linear combination of 0 and 0 , respectively.
0
1
e the differential operator on (0, 1) introduced in Corollary 9.1:
Let us consider D,
e = y(1 − y)H 00 (y) + 1 n(1 − 2y)H 0 (y)
DH
2


0
−`
0
0
2
(1 − 2y) + 1 
(1 − 2y)  `+1
0 −` − 1 0  H(y) +
+
− 2
4y(1 − y)
2y(1 − y)
0
0
−`
0

−`
0
 H(y).
0 − `+1
2
−`
0
Recall that the operator D that appears in Theorem 9.3 extends the differential operator D =
e −1 to the whole real line.
ΨDΨ
36
J.A. Tirao and I.N. Zurrián
We want to focus our attention on the following vector spaces of C3 -valued analytic functions
on (0, 1):
e = λH, H( cos s+1 ) analytic at s = 0 ,
Sλ = H = H(y) : DH
2
Wλ = P = P (y) : DP = λP, analytic on [0, 1] .
From Theorem 9.2 we know that the correspondence P 7→ ΨP is an injective linear map
from Wλ into Sλ . In fact, Ψ((cos s + 1)/2) is analytic as a function of s and P is analytic at
y = 1, hence H((cos s + 1)/2) = (ΨP )((cos s + 1)/2) is analytic at s = 0.
Then, we have an analogous result to Theorem 7.3, whose proof is quite similar and therefore
we will omit it.
Theorem 9.4. The linear map P 7→ ΨP is an isomorphism from Wλ onto Sλ .
Now, we can easily make a proof similar to that one of Theorem 7.4 in order to obtain next
theorem.
Theorem 9.5. Let H be the C3 -valued analytic function on (0, 1) given by an irreducible spherical function Φ on SO(2` + 2) of fundamental SO(2` + 1)-type (1, . . . , 1) ∈ C` . If P = Ψ−1 H,
then P is polynomial.
ˆ
For a given fundamental K-type π ∈ SO(n),
n = 2` + 1, with highest weight (1, . . . , 1) ∈ C` ,
let Φw,δ denote the irreducible spherical function of the pair (SO(n + 1), SO(n)) given by τ ∈
ˆ
SO(n
+ 1) with highest weight of the form (w + 1, 1, . . . , 1, δ) ∈ C`+1 , δ = −1, 0, 1.
Now, combining Theorems 9.3, 9.5 and the expression of the eigenvalue λn (w, δ) given in (6.2)
we have the following statement.
Theorem 9.6. Given w ∈ N, every irreducible spherical function Φw,δ of the pair (SO(n + 1),
SO(n)) with n = 2` + 1, of type mn = (1, . . . , 1) ∈ C` , corresponds to a vector-valued function Pw,δ (w ≥ 0, δ = −1,
0, 1), which is a polynomial of degree w. The leading coefficients
0
of Pw,0 is a multiple of 1 and the leading coefficients of Pw,−1 and Pw,1 are both linear com0
1
0
binations of 0 and 0 . Precisely
0
Pw,δ (y) =
1
w
X
yj
j=0
j!
[C; U ; V + λ]j Pw,δ (0),
with


(n + 2)/2
1/2
0
,
1
(n + 2)/2
1
C=
0
1/2
(n + 2)/2


−` − 1 0
0
−`
0 ,
U = (n + 2)I,
V = 0
0
0 −` − 1
(
−w(w + n + 1) − `
if δ = 0,
λ = λn (w, δ) =
−w(w + n + 1) − ` − 1 if δ = ±1.
Even more, the value of Pw,δ (0) can be computed.
Spherical Functions of Fundamental K-Types Associated with the n-Dimensional Sphere
37
Proof . It only remains to prove that Pw,δ (0) can be computed.
Let us consider the case δ = 0. We know from (6.2) and Theorem 9.3 that there is some
c ∈ C such that
 
0

[C; U ; V + λ]w Pw,0 (0) = c 1 .
0
Since [C; U ; V + λ]w is invertible this c is univocally determined by the condition Φ(e) = I which
implies
 
w
1
X
1
[C; U ; V + λ]j Pw,0 (0) = 1 .
Ψ(1)
j!
j=0
1
Now let us consider the cases δ = ±1. We know from (6.2) and Theorem 9.3 that
*1 0+
[C; U ; V + λ]w Pw,δ (0) ∈ 0 , 0 ;
0
1
since [C; U ; V + λ]w is invertible, this condition tells us that Pw,δ (0) belongs to a plane which
contains the origin and does not depend on δ.
Besides, the condition Φw,δ (e) = I, for δ = ±1, tells us
  

w
1
1 1 1 X
1
1 = 1 1 1
[C; U ; V + λ]j Pw,δ (0).
j!
1
1 1 1 j=0
Then, Pw,δ (0) belongs to a plane, parallel to the kernel of


w
1 1 1 X
1
1 1 1
[C; U ; V + λ]j ,
j!
1 1 1 j=0
which does not contain the origin and does not depend on δ. Therefore we know that both
Pw,1 (0) and Pw,−1 (0) are in the same straight line.
On the other hand, recall that we have
cos s + 1
cos s + 1
Φw,δ (a(s)) = Ψ
Pw,δ
,
2
2
where


In−1
0
0
cos s sin s  ,
a(s) =  0
0
− sin s cos s
then


i 0 −i
d
Φ(a(s)) =  0 0 0  Pw,δ (1).
ds s=0
−i 0 i
38
J.A. Tirao and I.N. Zurrián
d
From [24, p. 364, equation (8)] we can easily compute ds
Φw,δ (a(s)) at s = 0, which is obtained
by looking at the action of τ̇ (In+1,n ) and considering the corresponding projection, see (2.1);
having then
  

w
−1
i 0 −i X
1
i(w + ` + 1)   

0
= 0 0 0
[C; U ; V + λ]j Pw,δ (0).
δ
1+`
j!
1
−i 0 i j=0
This last condition establishes that Pw,1 (0) and Pw,−1 (0) are in two different and parallel
planes, and the line mentioned above does not belong to any of them since each plane has to
intersect it. Therefore the values of Pw,1 (0) and Pw,−1 (0) are univocally determined.
9.2
Matrix-valued orthogonal polynomials of size 3
In this subsection, given n of the form 2` + 1 with ` ∈ N, we shall construct a sequence of
matrix-valued polynomials {Pw }w≥0 directly related to irreducible spherical functions of type
ˆ
π ∈ SO(n)
of highest weight mπ = (1, . . . , 1) ∈ C` .
Given a nonnegative integer w and δ = −1, 0, 1, we can consider Φw,δ , the irreducible spherical
ˆ
function of type π associated with the irreducible representation τ ∈ SO(n
+ 1) of highest weight
of the form mτ = (w + 1, 1, . . . , 1, δ).
We insist on recalling that, since π has only three SO(2`)-submodules, we can interpret the
diagonal matrix-valued function Φw,δ (a(s)), s ∈ (0, π), as a 3 column vector function.
Now we consider the vector-valued function
Pw,δ : (0, 1) → C3
given by the vector function Pw,δ (y) = Ψ−1 (y)Φw,δ (a(s)), with cos(s) = 2y − 1. Then, we define
the matrix-valued function
Pw = Pw (y),
whose δ-th column (δ = −1, 0, 1) is given by the C3 -valued polynomial Pw,δ (y).
Let consider the matrix-valued skew symmetric bilinear form defined among continuous 3 × 3
matrix-valued functions on [0, 1] by
Z 1
Q∗ (y)W (y)P (y)dy,
hP, QiW =
0
where the 3 × 3 weight-matrix W is given by


d1 0 0
(n − 1)!!
W (y) =
(y(1 − y))n/2−1 Ψ∗ (y)  0 d2 0  Ψ(y)
(n − 2)!!
0 0 d3
with
d1 = d3 =
(2` + 1)!
,
`!(` + 2)!
d2 =
(2` + 1)!
,
`!(` + 1)!
and
p
p


2y − 1 + 2i y − y 2
1
2y − 1 − 2i y − y 2

1p
2y − 1
1p
Ψ(y) = 
2
2
2y − 1 − 2i y − y
1
2y − 1 + 2i y − y
Let us recall that, from Proposition 7.1, we have
Z 1
∗
0
0
hΦw,δ , Φw ,δ i =
Pw,δ
W (y)Pw0 ,δ0 dy.
0
Spherical Functions of Fundamental K-Types Associated with the n-Dimensional Sphere
Remark 9.7. Notice that W reduces to a smaller size: if M =
1 √0 1
0
20
−1 0 1
39
we have
(n − 1)!!
M W (y)M ∗ =
(y(1 − y))n/2−1 4
(n − 2)!!
√


2
0
2d1 (2y
√ d1 (2y − 1) + d2 (2y −2 1)/ 2
√ − 1) + d2
.
× d1 (2y − 1) 2 + d2 (2y − 1)/ 2
d1 + d2 (2y − 1) /2
0
2
0
0
d1 8(y − y )
Then we state the following theorem.
Theorem 9.8. The matrix-valued polynomial functions Pw , w ≥ 0, form a sequence of orthogonal polynomials with respect to W , which are eigenfunctions of the symmetric differential
operator D from Theorem 9.2. Moreover,


λ(w, −1)
0
0
0
λ(w, 0)
0 ,
DPw = Pw 
0
0
λ(w, 1)
where
(
−w(w + n + 1) − p
λ(w, δ) =
−w(w + n + 1) − n + p
if
if
δ = 0,
δ = ±1.
Proof . The proof is completely analogous to the proof of Theorem 8.1
Appendix
Proof of Proposition 3.2. For |t| sufficiently small A(s, t) is close to the identity of K, i.e. to
the identity matrix In . So we can consider the function
X(s, t) = log(A(s, t)) = B(s, t) −
B(s, t)2 B(s, t)3
+
− ··· ,
2
3
(9.2)
where B(s, t) = A(s, t) − In . Then
π(A(s, t)) = π(exp X(s, t)) = exp π̇(X(s, t)) =
X π̇(X(s, t))j
j≥0
j!
.
Now we differentiate with respect to t to obtain
∂(π ◦ A)
∂X
∂X
∂X
1
1
= π̇
+ 2! π̇
π̇(X) + 2! π̇(X)π̇
∂t
∂t
∂t
∂t
∂X
∂X
∂X
+ 3!1 π̇
π̇(X)2 + 3!1 π̇(X)π̇
π̇ (X) + 3!1 π̇(X)2 π̇
+ · · · . (9.3)
∂t
∂t
∂t
Since X(s, 0) = 0, if we differentiate (9.2) with respect to t and evaluate at (s, 0) we obtain
2
2
∂ 2 (π ◦ A)
∂ X
∂X
= π̇
+ π̇
.
∂t2
∂t2 t=0
∂t t=0
t=0
∂X
∂t t=0
2
and ∂∂tX
2 t=0 we differentiate (9.2) and we get
∂X
∂B 1 ∂B
∂B
∂B
2
1
1 ∂B
1
1 2 ∂B
=
−2
B − 2B
+3
B + 3B
B + 3B
+ ··· .
∂t
∂t
∂t
∂t
∂t
∂t
∂t
To compute
40
J.A. Tirao and I.N. Zurrián
Since B(s, 0) = 0 we have
∂X
∂t
t=0
=
∂B
∂t
t=0
=
∂A
∂t
t=0
.
We also get
∂2X
∂t2
t=0
=
∂2A
∂t2
t=0
−
2
∂A
∂t
t=0
.
Now we will first consider the case A(s, t) = k(s, t). A direct computation yields to

0
0
∂k 

= 0

∂t
0
0
in particular ∂k
∂t
t = 0 we obtain
∂2A
∂t2
0
− sin s sin t
(1−cos2 s cos2 t)3/2
0
− sin2 s cos t
(1−cos2 s cos2 t)3/2
0
0
sin2 s cos t
(1−cos2 s cos2 t)3/2
0
− sin s sin t
(1−cos2 s cos2 t)3/2
0

0
0


0 ,

0
0
= sin1 s In,j . Differentiating once more with
t=0
2
∂ k
= − sin12 s (Ejj + En,n ). Then we get
∂t2 t=0
t=0
0
0
0
0
0
−
∂A
∂t
2
=−
t=0
respect to t and evaluating at
1
1
(Ejj + En,n ) −
I 2 = 0.
sin2 s
sin2 s n,j
Similarly when A(s, t) = h(s, t) we obtain
0
0
∂h 

= 0

∂t
0
0
0

in particular ∂h
∂t
t = 0 we obtain
∂2A
∂t2
− sin s cos2 s cos t sin t
(1−cos2 s cos2 t)3/2
0
cos s cos t sin2 s
(1−cos2 s cos2 t)3/2
0
t=0
0
− cos s cos t sin2 s
(1−cos2 s cos2 t)3/2
0
− sin s cos2 s cos t sin t
(1−cos2 s cos2 t)3/2
0

0
0


0 ,

0
0
s
= − cos
sin s In,j . Differentiating once more with
t=0
2
2s
∂ h
(Ejj + En,n ). Then we get
= − cos
∂t2 t=0
sin2 s
−
0
0
0
0
0
∂A
∂t
2
t=0
=−
respect to t and evaluating at
cos2 s
cos2 s 2
I = 0.
2 (Ejj + En,n ) −
sin s
sin2 s n,j
Proposition follows.
Acknowledgements
This paper was partially supported by CONICET, PIP 112-200801-01533 and SeCyT-UNC.
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